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## G = C2×C22⋊S4order 192 = 26·3

### Direct product of C2 and C22⋊S4

Aliases: C2×C22⋊S4, C233S4, C253S3, C245D6, C22⋊(C2×S4), C22⋊A44C22, (C2×C22⋊A4)⋊2C2, SmallGroup(192,1538)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C22⋊A4 — C2×C22⋊S4
 Chief series C1 — C22 — C24 — C22⋊A4 — C22⋊S4 — C2×C22⋊S4
 Lower central C22⋊A4 — C2×C22⋊S4
 Upper central C1 — C2

Generators and relations for C2×C22⋊S4
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fcf-1=bc=cb, bd=db, be=eb, fbf-1=gbg=c, cd=dc, ce=ec, gcg=b, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 1120 in 238 conjugacy classes, 15 normal (7 characteristic)
C1, C2, C2 [×10], C3, C4 [×6], C22 [×3], C22 [×44], S3 [×2], C6, C2×C4 [×12], D4 [×24], C23 [×3], C23 [×42], A4 [×4], D6, C22⋊C4 [×12], C22×C4 [×3], C2×D4 [×24], C24, C24 [×9], S4 [×6], C2×A4 [×4], C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4 [×3], C25, C2×S4 [×3], C22⋊A4, C2×C22≀C2, C22⋊S4 [×2], C2×C22⋊A4, C2×C22⋊S4
Quotients: C1, C2 [×3], C22, S3, D6, S4 [×3], C2×S4 [×3], C22⋊S4, C2×C22⋊S4

Character table of C2×C22⋊S4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 6 size 1 1 3 3 3 3 3 3 6 6 12 12 32 12 12 12 12 12 12 32 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 linear of order 2 ρ4 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ5 2 2 2 2 2 2 2 2 2 2 0 0 -1 0 0 0 0 0 0 -1 orthogonal lifted from S3 ρ6 2 -2 -2 2 2 -2 -2 2 2 -2 0 0 -1 0 0 0 0 0 0 1 orthogonal lifted from D6 ρ7 3 -3 1 -1 3 -3 1 -1 -1 1 -1 1 0 -1 1 -1 1 -1 1 0 orthogonal lifted from C2×S4 ρ8 3 -3 -3 3 -1 1 1 -1 -1 1 1 -1 0 -1 1 -1 1 1 -1 0 orthogonal lifted from C2×S4 ρ9 3 3 -1 -1 -1 -1 3 3 -1 -1 1 1 0 -1 -1 -1 -1 1 1 0 orthogonal lifted from S4 ρ10 3 -3 1 -1 -1 1 -3 3 -1 1 -1 1 0 -1 1 1 -1 1 -1 0 orthogonal lifted from C2×S4 ρ11 3 3 -1 -1 3 3 -1 -1 -1 -1 1 1 0 -1 -1 1 1 -1 -1 0 orthogonal lifted from S4 ρ12 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 0 -1 -1 1 1 1 1 0 orthogonal lifted from S4 ρ13 3 3 -1 -1 -1 -1 3 3 -1 -1 -1 -1 0 1 1 1 1 -1 -1 0 orthogonal lifted from S4 ρ14 3 -3 -3 3 -1 1 1 -1 -1 1 -1 1 0 1 -1 1 -1 -1 1 0 orthogonal lifted from C2×S4 ρ15 3 -3 1 -1 3 -3 1 -1 -1 1 1 -1 0 1 -1 1 -1 1 -1 0 orthogonal lifted from C2×S4 ρ16 3 3 3 3 -1 -1 -1 -1 -1 -1 1 1 0 1 1 -1 -1 -1 -1 0 orthogonal lifted from S4 ρ17 3 3 -1 -1 3 3 -1 -1 -1 -1 -1 -1 0 1 1 -1 -1 1 1 0 orthogonal lifted from S4 ρ18 3 -3 1 -1 -1 1 -3 3 -1 1 1 -1 0 1 -1 -1 1 -1 1 0 orthogonal lifted from C2×S4 ρ19 6 6 -2 -2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C22⋊S4 ρ20 6 -6 2 -2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C2×C22⋊S4
On 12 points - transitive group 12T100
Generators in S12
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)
(1 8)(2 5)(3 12)(4 10)(6 7)(9 11)
(1 4)(2 11)(3 7)(5 9)(6 12)(8 10)
(1 10)(2 11)(4 8)(5 9)
(2 11)(3 12)(5 9)(6 7)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(2 3)(4 8)(5 7)(6 9)(11 12)

G:=sub<Sym(12)| (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (1,8)(2,5)(3,12)(4,10)(6,7)(9,11), (1,4)(2,11)(3,7)(5,9)(6,12)(8,10), (1,10)(2,11)(4,8)(5,9), (2,11)(3,12)(5,9)(6,7), (1,2,3)(4,5,6)(7,8,9)(10,11,12), (2,3)(4,8)(5,7)(6,9)(11,12)>;

G:=Group( (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (1,8)(2,5)(3,12)(4,10)(6,7)(9,11), (1,4)(2,11)(3,7)(5,9)(6,12)(8,10), (1,10)(2,11)(4,8)(5,9), (2,11)(3,12)(5,9)(6,7), (1,2,3)(4,5,6)(7,8,9)(10,11,12), (2,3)(4,8)(5,7)(6,9)(11,12) );

G=PermutationGroup([(1,10),(2,11),(3,12),(4,8),(5,9),(6,7)], [(1,8),(2,5),(3,12),(4,10),(6,7),(9,11)], [(1,4),(2,11),(3,7),(5,9),(6,12),(8,10)], [(1,10),(2,11),(4,8),(5,9)], [(2,11),(3,12),(5,9),(6,7)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(2,3),(4,8),(5,7),(6,9),(11,12)])

G:=TransitiveGroup(12,100);

On 12 points - transitive group 12T101
Generators in S12
(1 10)(2 11)(3 12)(4 8)(5 9)(6 7)
(1 4)(2 5)(8 10)(9 11)
(1 4)(3 6)(7 12)(8 10)
(1 8)(2 9)(4 10)(5 11)
(2 9)(3 7)(5 11)(6 12)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(2 3)(5 6)(7 9)(11 12)

G:=sub<Sym(12)| (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (1,4)(2,5)(8,10)(9,11), (1,4)(3,6)(7,12)(8,10), (1,8)(2,9)(4,10)(5,11), (2,9)(3,7)(5,11)(6,12), (1,2,3)(4,5,6)(7,8,9)(10,11,12), (2,3)(5,6)(7,9)(11,12)>;

G:=Group( (1,10)(2,11)(3,12)(4,8)(5,9)(6,7), (1,4)(2,5)(8,10)(9,11), (1,4)(3,6)(7,12)(8,10), (1,8)(2,9)(4,10)(5,11), (2,9)(3,7)(5,11)(6,12), (1,2,3)(4,5,6)(7,8,9)(10,11,12), (2,3)(5,6)(7,9)(11,12) );

G=PermutationGroup([(1,10),(2,11),(3,12),(4,8),(5,9),(6,7)], [(1,4),(2,5),(8,10),(9,11)], [(1,4),(3,6),(7,12),(8,10)], [(1,8),(2,9),(4,10),(5,11)], [(2,9),(3,7),(5,11),(6,12)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(2,3),(5,6),(7,9),(11,12)])

G:=TransitiveGroup(12,101);

On 12 points - transitive group 12T103
Generators in S12
(1 4)(2 5)(3 6)(7 12)(8 10)(9 11)
(1 4)(2 5)(8 10)(9 11)
(1 4)(3 6)(7 12)(8 10)
(1 8)(2 9)(4 10)(5 11)
(2 9)(3 7)(5 11)(6 12)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 10)(2 12)(3 11)(4 8)(5 7)(6 9)

G:=sub<Sym(12)| (1,4)(2,5)(3,6)(7,12)(8,10)(9,11), (1,4)(2,5)(8,10)(9,11), (1,4)(3,6)(7,12)(8,10), (1,8)(2,9)(4,10)(5,11), (2,9)(3,7)(5,11)(6,12), (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,10)(2,12)(3,11)(4,8)(5,7)(6,9)>;

G:=Group( (1,4)(2,5)(3,6)(7,12)(8,10)(9,11), (1,4)(2,5)(8,10)(9,11), (1,4)(3,6)(7,12)(8,10), (1,8)(2,9)(4,10)(5,11), (2,9)(3,7)(5,11)(6,12), (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,10)(2,12)(3,11)(4,8)(5,7)(6,9) );

G=PermutationGroup([(1,4),(2,5),(3,6),(7,12),(8,10),(9,11)], [(1,4),(2,5),(8,10),(9,11)], [(1,4),(3,6),(7,12),(8,10)], [(1,8),(2,9),(4,10),(5,11)], [(2,9),(3,7),(5,11),(6,12)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(1,10),(2,12),(3,11),(4,8),(5,7),(6,9)])

G:=TransitiveGroup(12,103);

On 12 points - transitive group 12T106
Generators in S12
(1 7)(2 8)(3 9)(4 12)(5 10)(6 11)
(1 7)(3 9)(4 12)(5 10)
(2 8)(3 9)(4 12)(6 11)
(1 7)(2 8)(4 12)(6 11)
(2 8)(3 9)(4 12)(5 10)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 11)(2 10)(3 12)(4 9)(5 8)(6 7)

G:=sub<Sym(12)| (1,7)(2,8)(3,9)(4,12)(5,10)(6,11), (1,7)(3,9)(4,12)(5,10), (2,8)(3,9)(4,12)(6,11), (1,7)(2,8)(4,12)(6,11), (2,8)(3,9)(4,12)(5,10), (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,11)(2,10)(3,12)(4,9)(5,8)(6,7)>;

G:=Group( (1,7)(2,8)(3,9)(4,12)(5,10)(6,11), (1,7)(3,9)(4,12)(5,10), (2,8)(3,9)(4,12)(6,11), (1,7)(2,8)(4,12)(6,11), (2,8)(3,9)(4,12)(5,10), (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,11)(2,10)(3,12)(4,9)(5,8)(6,7) );

G=PermutationGroup([(1,7),(2,8),(3,9),(4,12),(5,10),(6,11)], [(1,7),(3,9),(4,12),(5,10)], [(2,8),(3,9),(4,12),(6,11)], [(1,7),(2,8),(4,12),(6,11)], [(2,8),(3,9),(4,12),(5,10)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(1,11),(2,10),(3,12),(4,9),(5,8),(6,7)])

G:=TransitiveGroup(12,106);

On 16 points - transitive group 16T429
Generators in S16
(1 2)(3 4)(5 11)(6 12)(7 13)(8 14)(9 15)(10 16)
(1 12)(2 6)(3 9)(4 15)(5 7)(8 10)(11 13)(14 16)
(1 11)(2 5)(3 8)(4 14)(6 7)(9 10)(12 13)(15 16)
(1 13)(2 7)(3 8)(4 14)(5 6)(9 10)(11 12)(15 16)
(1 11)(2 5)(3 9)(4 15)(6 7)(8 10)(12 13)(14 16)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
(1 3)(2 4)(5 15)(6 14)(7 16)(8 12)(9 11)(10 13)

G:=sub<Sym(16)| (1,2)(3,4)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16), (1,12)(2,6)(3,9)(4,15)(5,7)(8,10)(11,13)(14,16), (1,11)(2,5)(3,8)(4,14)(6,7)(9,10)(12,13)(15,16), (1,13)(2,7)(3,8)(4,14)(5,6)(9,10)(11,12)(15,16), (1,11)(2,5)(3,9)(4,15)(6,7)(8,10)(12,13)(14,16), (5,6,7)(8,9,10)(11,12,13)(14,15,16), (1,3)(2,4)(5,15)(6,14)(7,16)(8,12)(9,11)(10,13)>;

G:=Group( (1,2)(3,4)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16), (1,12)(2,6)(3,9)(4,15)(5,7)(8,10)(11,13)(14,16), (1,11)(2,5)(3,8)(4,14)(6,7)(9,10)(12,13)(15,16), (1,13)(2,7)(3,8)(4,14)(5,6)(9,10)(11,12)(15,16), (1,11)(2,5)(3,9)(4,15)(6,7)(8,10)(12,13)(14,16), (5,6,7)(8,9,10)(11,12,13)(14,15,16), (1,3)(2,4)(5,15)(6,14)(7,16)(8,12)(9,11)(10,13) );

G=PermutationGroup([(1,2),(3,4),(5,11),(6,12),(7,13),(8,14),(9,15),(10,16)], [(1,12),(2,6),(3,9),(4,15),(5,7),(8,10),(11,13),(14,16)], [(1,11),(2,5),(3,8),(4,14),(6,7),(9,10),(12,13),(15,16)], [(1,13),(2,7),(3,8),(4,14),(5,6),(9,10),(11,12),(15,16)], [(1,11),(2,5),(3,9),(4,15),(6,7),(8,10),(12,13),(14,16)], [(5,6,7),(8,9,10),(11,12,13),(14,15,16)], [(1,3),(2,4),(5,15),(6,14),(7,16),(8,12),(9,11),(10,13)])

G:=TransitiveGroup(16,429);

On 24 points - transitive group 24T432
Generators in S24
(1 15)(2 13)(3 14)(4 24)(5 22)(6 23)(7 16)(8 17)(9 18)(10 21)(11 19)(12 20)
(1 11)(3 10)(4 17)(5 18)(8 24)(9 22)(14 21)(15 19)
(2 12)(3 10)(4 17)(6 16)(7 23)(8 24)(13 20)(14 21)
(2 20)(3 21)(5 9)(6 7)(10 14)(12 13)(16 23)(18 22)
(1 19)(3 21)(4 8)(6 7)(10 14)(11 15)(16 23)(17 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 6)(2 5)(3 4)(7 19)(8 21)(9 20)(10 17)(11 16)(12 18)(13 22)(14 24)(15 23)

G:=sub<Sym(24)| (1,15)(2,13)(3,14)(4,24)(5,22)(6,23)(7,16)(8,17)(9,18)(10,21)(11,19)(12,20), (1,11)(3,10)(4,17)(5,18)(8,24)(9,22)(14,21)(15,19), (2,12)(3,10)(4,17)(6,16)(7,23)(8,24)(13,20)(14,21), (2,20)(3,21)(5,9)(6,7)(10,14)(12,13)(16,23)(18,22), (1,19)(3,21)(4,8)(6,7)(10,14)(11,15)(16,23)(17,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23)>;

G:=Group( (1,15)(2,13)(3,14)(4,24)(5,22)(6,23)(7,16)(8,17)(9,18)(10,21)(11,19)(12,20), (1,11)(3,10)(4,17)(5,18)(8,24)(9,22)(14,21)(15,19), (2,12)(3,10)(4,17)(6,16)(7,23)(8,24)(13,20)(14,21), (2,20)(3,21)(5,9)(6,7)(10,14)(12,13)(16,23)(18,22), (1,19)(3,21)(4,8)(6,7)(10,14)(11,15)(16,23)(17,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23) );

G=PermutationGroup([(1,15),(2,13),(3,14),(4,24),(5,22),(6,23),(7,16),(8,17),(9,18),(10,21),(11,19),(12,20)], [(1,11),(3,10),(4,17),(5,18),(8,24),(9,22),(14,21),(15,19)], [(2,12),(3,10),(4,17),(6,16),(7,23),(8,24),(13,20),(14,21)], [(2,20),(3,21),(5,9),(6,7),(10,14),(12,13),(16,23),(18,22)], [(1,19),(3,21),(4,8),(6,7),(10,14),(11,15),(16,23),(17,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,6),(2,5),(3,4),(7,19),(8,21),(9,20),(10,17),(11,16),(12,18),(13,22),(14,24),(15,23)])

G:=TransitiveGroup(24,432);

On 24 points - transitive group 24T485
Generators in S24
(1 19)(2 20)(3 21)(4 8)(5 9)(6 7)(10 14)(11 15)(12 13)(16 23)(17 24)(18 22)
(1 19)(3 21)(4 8)(5 9)(10 14)(11 15)(17 24)(18 22)
(2 20)(3 21)(4 8)(6 7)(10 14)(12 13)(16 23)(17 24)
(1 15)(2 20)(3 10)(4 24)(5 18)(6 7)(8 17)(9 22)(11 19)(12 13)(14 21)(16 23)
(1 11)(2 13)(3 21)(4 8)(5 22)(6 16)(7 23)(9 18)(10 14)(12 20)(15 19)(17 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 6)(2 5)(3 4)(7 19)(8 21)(9 20)(10 17)(11 16)(12 18)(13 22)(14 24)(15 23)

G:=sub<Sym(24)| (1,19)(2,20)(3,21)(4,8)(5,9)(6,7)(10,14)(11,15)(12,13)(16,23)(17,24)(18,22), (1,19)(3,21)(4,8)(5,9)(10,14)(11,15)(17,24)(18,22), (2,20)(3,21)(4,8)(6,7)(10,14)(12,13)(16,23)(17,24), (1,15)(2,20)(3,10)(4,24)(5,18)(6,7)(8,17)(9,22)(11,19)(12,13)(14,21)(16,23), (1,11)(2,13)(3,21)(4,8)(5,22)(6,16)(7,23)(9,18)(10,14)(12,20)(15,19)(17,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23)>;

G:=Group( (1,19)(2,20)(3,21)(4,8)(5,9)(6,7)(10,14)(11,15)(12,13)(16,23)(17,24)(18,22), (1,19)(3,21)(4,8)(5,9)(10,14)(11,15)(17,24)(18,22), (2,20)(3,21)(4,8)(6,7)(10,14)(12,13)(16,23)(17,24), (1,15)(2,20)(3,10)(4,24)(5,18)(6,7)(8,17)(9,22)(11,19)(12,13)(14,21)(16,23), (1,11)(2,13)(3,21)(4,8)(5,22)(6,16)(7,23)(9,18)(10,14)(12,20)(15,19)(17,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23) );

G=PermutationGroup([(1,19),(2,20),(3,21),(4,8),(5,9),(6,7),(10,14),(11,15),(12,13),(16,23),(17,24),(18,22)], [(1,19),(3,21),(4,8),(5,9),(10,14),(11,15),(17,24),(18,22)], [(2,20),(3,21),(4,8),(6,7),(10,14),(12,13),(16,23),(17,24)], [(1,15),(2,20),(3,10),(4,24),(5,18),(6,7),(8,17),(9,22),(11,19),(12,13),(14,21),(16,23)], [(1,11),(2,13),(3,21),(4,8),(5,22),(6,16),(7,23),(9,18),(10,14),(12,20),(15,19),(17,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,6),(2,5),(3,4),(7,19),(8,21),(9,20),(10,17),(11,16),(12,18),(13,22),(14,24),(15,23)])

G:=TransitiveGroup(24,485);

On 24 points - transitive group 24T486
Generators in S24
(1 16)(2 17)(3 18)(4 12)(5 10)(6 11)(7 13)(8 14)(9 15)(19 23)(20 24)(21 22)
(1 4)(2 9)(3 22)(5 24)(6 7)(8 23)(10 20)(11 13)(12 16)(14 19)(15 17)(18 21)
(1 8)(2 24)(3 6)(4 23)(5 9)(7 22)(10 15)(11 18)(12 19)(13 21)(14 16)(17 20)
(1 16)(2 17)(4 12)(5 10)(8 14)(9 15)(19 23)(20 24)
(2 17)(3 18)(5 10)(6 11)(7 13)(9 15)(20 24)(21 22)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(2 3)(4 8)(5 7)(6 9)(10 13)(11 15)(12 14)(17 18)(20 21)(22 24)

G:=sub<Sym(24)| (1,16)(2,17)(3,18)(4,12)(5,10)(6,11)(7,13)(8,14)(9,15)(19,23)(20,24)(21,22), (1,4)(2,9)(3,22)(5,24)(6,7)(8,23)(10,20)(11,13)(12,16)(14,19)(15,17)(18,21), (1,8)(2,24)(3,6)(4,23)(5,9)(7,22)(10,15)(11,18)(12,19)(13,21)(14,16)(17,20), (1,16)(2,17)(4,12)(5,10)(8,14)(9,15)(19,23)(20,24), (2,17)(3,18)(5,10)(6,11)(7,13)(9,15)(20,24)(21,22), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (2,3)(4,8)(5,7)(6,9)(10,13)(11,15)(12,14)(17,18)(20,21)(22,24)>;

G:=Group( (1,16)(2,17)(3,18)(4,12)(5,10)(6,11)(7,13)(8,14)(9,15)(19,23)(20,24)(21,22), (1,4)(2,9)(3,22)(5,24)(6,7)(8,23)(10,20)(11,13)(12,16)(14,19)(15,17)(18,21), (1,8)(2,24)(3,6)(4,23)(5,9)(7,22)(10,15)(11,18)(12,19)(13,21)(14,16)(17,20), (1,16)(2,17)(4,12)(5,10)(8,14)(9,15)(19,23)(20,24), (2,17)(3,18)(5,10)(6,11)(7,13)(9,15)(20,24)(21,22), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (2,3)(4,8)(5,7)(6,9)(10,13)(11,15)(12,14)(17,18)(20,21)(22,24) );

G=PermutationGroup([(1,16),(2,17),(3,18),(4,12),(5,10),(6,11),(7,13),(8,14),(9,15),(19,23),(20,24),(21,22)], [(1,4),(2,9),(3,22),(5,24),(6,7),(8,23),(10,20),(11,13),(12,16),(14,19),(15,17),(18,21)], [(1,8),(2,24),(3,6),(4,23),(5,9),(7,22),(10,15),(11,18),(12,19),(13,21),(14,16),(17,20)], [(1,16),(2,17),(4,12),(5,10),(8,14),(9,15),(19,23),(20,24)], [(2,17),(3,18),(5,10),(6,11),(7,13),(9,15),(20,24),(21,22)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(2,3),(4,8),(5,7),(6,9),(10,13),(11,15),(12,14),(17,18),(20,21),(22,24)])

G:=TransitiveGroup(24,486);

On 24 points - transitive group 24T487
Generators in S24
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)
(1 18)(2 13)(3 12)(4 8)(5 21)(6 24)(7 19)(9 23)(10 15)(11 16)(14 17)(20 22)
(1 15)(2 11)(3 17)(4 20)(5 23)(6 7)(8 22)(9 21)(10 18)(12 14)(13 16)(19 24)
(1 10)(2 11)(4 22)(5 23)(8 20)(9 21)(13 16)(15 18)
(2 11)(3 12)(5 23)(6 24)(7 19)(9 21)(13 16)(14 17)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 4)(2 6)(3 5)(7 13)(8 15)(9 14)(10 22)(11 24)(12 23)(16 19)(17 21)(18 20)

G:=sub<Sym(24)| (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,18)(2,13)(3,12)(4,8)(5,21)(6,24)(7,19)(9,23)(10,15)(11,16)(14,17)(20,22), (1,15)(2,11)(3,17)(4,20)(5,23)(6,7)(8,22)(9,21)(10,18)(12,14)(13,16)(19,24), (1,10)(2,11)(4,22)(5,23)(8,20)(9,21)(13,16)(15,18), (2,11)(3,12)(5,23)(6,24)(7,19)(9,21)(13,16)(14,17), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,4)(2,6)(3,5)(7,13)(8,15)(9,14)(10,22)(11,24)(12,23)(16,19)(17,21)(18,20)>;

G:=Group( (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,18)(2,13)(3,12)(4,8)(5,21)(6,24)(7,19)(9,23)(10,15)(11,16)(14,17)(20,22), (1,15)(2,11)(3,17)(4,20)(5,23)(6,7)(8,22)(9,21)(10,18)(12,14)(13,16)(19,24), (1,10)(2,11)(4,22)(5,23)(8,20)(9,21)(13,16)(15,18), (2,11)(3,12)(5,23)(6,24)(7,19)(9,21)(13,16)(14,17), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,4)(2,6)(3,5)(7,13)(8,15)(9,14)(10,22)(11,24)(12,23)(16,19)(17,21)(18,20) );

G=PermutationGroup([(1,4),(2,5),(3,6),(7,17),(8,18),(9,16),(10,22),(11,23),(12,24),(13,21),(14,19),(15,20)], [(1,18),(2,13),(3,12),(4,8),(5,21),(6,24),(7,19),(9,23),(10,15),(11,16),(14,17),(20,22)], [(1,15),(2,11),(3,17),(4,20),(5,23),(6,7),(8,22),(9,21),(10,18),(12,14),(13,16),(19,24)], [(1,10),(2,11),(4,22),(5,23),(8,20),(9,21),(13,16),(15,18)], [(2,11),(3,12),(5,23),(6,24),(7,19),(9,21),(13,16),(14,17)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,4),(2,6),(3,5),(7,13),(8,15),(9,14),(10,22),(11,24),(12,23),(16,19),(17,21),(18,20)])

G:=TransitiveGroup(24,487);

On 24 points - transitive group 24T488
Generators in S24
(1 8)(2 9)(3 7)(4 23)(5 24)(6 22)(10 20)(11 21)(12 19)(13 18)(14 16)(15 17)
(1 19)(2 10)(3 7)(4 14)(5 17)(6 22)(8 12)(9 20)(11 21)(13 18)(15 24)(16 23)
(1 12)(2 9)(3 21)(4 16)(5 24)(6 13)(7 11)(8 19)(10 20)(14 23)(15 17)(18 22)
(1 16)(2 15)(3 7)(4 12)(5 20)(6 22)(8 14)(9 17)(10 24)(11 21)(13 18)(19 23)
(1 8)(2 17)(3 13)(4 23)(5 10)(6 21)(7 18)(9 15)(11 22)(12 19)(14 16)(20 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(2 3)(5 6)(7 9)(10 21)(11 20)(12 19)(13 17)(14 16)(15 18)(22 24)

G:=sub<Sym(24)| (1,8)(2,9)(3,7)(4,23)(5,24)(6,22)(10,20)(11,21)(12,19)(13,18)(14,16)(15,17), (1,19)(2,10)(3,7)(4,14)(5,17)(6,22)(8,12)(9,20)(11,21)(13,18)(15,24)(16,23), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (1,16)(2,15)(3,7)(4,12)(5,20)(6,22)(8,14)(9,17)(10,24)(11,21)(13,18)(19,23), (1,8)(2,17)(3,13)(4,23)(5,10)(6,21)(7,18)(9,15)(11,22)(12,19)(14,16)(20,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (2,3)(5,6)(7,9)(10,21)(11,20)(12,19)(13,17)(14,16)(15,18)(22,24)>;

G:=Group( (1,8)(2,9)(3,7)(4,23)(5,24)(6,22)(10,20)(11,21)(12,19)(13,18)(14,16)(15,17), (1,19)(2,10)(3,7)(4,14)(5,17)(6,22)(8,12)(9,20)(11,21)(13,18)(15,24)(16,23), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (1,16)(2,15)(3,7)(4,12)(5,20)(6,22)(8,14)(9,17)(10,24)(11,21)(13,18)(19,23), (1,8)(2,17)(3,13)(4,23)(5,10)(6,21)(7,18)(9,15)(11,22)(12,19)(14,16)(20,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (2,3)(5,6)(7,9)(10,21)(11,20)(12,19)(13,17)(14,16)(15,18)(22,24) );

G=PermutationGroup([(1,8),(2,9),(3,7),(4,23),(5,24),(6,22),(10,20),(11,21),(12,19),(13,18),(14,16),(15,17)], [(1,19),(2,10),(3,7),(4,14),(5,17),(6,22),(8,12),(9,20),(11,21),(13,18),(15,24),(16,23)], [(1,12),(2,9),(3,21),(4,16),(5,24),(6,13),(7,11),(8,19),(10,20),(14,23),(15,17),(18,22)], [(1,16),(2,15),(3,7),(4,12),(5,20),(6,22),(8,14),(9,17),(10,24),(11,21),(13,18),(19,23)], [(1,8),(2,17),(3,13),(4,23),(5,10),(6,21),(7,18),(9,15),(11,22),(12,19),(14,16),(20,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(2,3),(5,6),(7,9),(10,21),(11,20),(12,19),(13,17),(14,16),(15,18),(22,24)])

G:=TransitiveGroup(24,488);

On 24 points - transitive group 24T489
Generators in S24
(1 8)(2 9)(3 7)(4 23)(5 24)(6 22)(10 20)(11 21)(12 19)(13 18)(14 16)(15 17)
(1 19)(2 10)(3 7)(4 14)(5 17)(6 22)(8 12)(9 20)(11 21)(13 18)(15 24)(16 23)
(1 12)(2 9)(3 21)(4 16)(5 24)(6 13)(7 11)(8 19)(10 20)(14 23)(15 17)(18 22)
(1 16)(2 15)(3 7)(4 12)(5 20)(6 22)(8 14)(9 17)(10 24)(11 21)(13 18)(19 23)
(1 8)(2 17)(3 13)(4 23)(5 10)(6 21)(7 18)(9 15)(11 22)(12 19)(14 16)(20 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 23)(2 22)(3 24)(4 8)(5 7)(6 9)(10 18)(11 17)(12 16)(13 20)(14 19)(15 21)

G:=sub<Sym(24)| (1,8)(2,9)(3,7)(4,23)(5,24)(6,22)(10,20)(11,21)(12,19)(13,18)(14,16)(15,17), (1,19)(2,10)(3,7)(4,14)(5,17)(6,22)(8,12)(9,20)(11,21)(13,18)(15,24)(16,23), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (1,16)(2,15)(3,7)(4,12)(5,20)(6,22)(8,14)(9,17)(10,24)(11,21)(13,18)(19,23), (1,8)(2,17)(3,13)(4,23)(5,10)(6,21)(7,18)(9,15)(11,22)(12,19)(14,16)(20,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,23)(2,22)(3,24)(4,8)(5,7)(6,9)(10,18)(11,17)(12,16)(13,20)(14,19)(15,21)>;

G:=Group( (1,8)(2,9)(3,7)(4,23)(5,24)(6,22)(10,20)(11,21)(12,19)(13,18)(14,16)(15,17), (1,19)(2,10)(3,7)(4,14)(5,17)(6,22)(8,12)(9,20)(11,21)(13,18)(15,24)(16,23), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (1,16)(2,15)(3,7)(4,12)(5,20)(6,22)(8,14)(9,17)(10,24)(11,21)(13,18)(19,23), (1,8)(2,17)(3,13)(4,23)(5,10)(6,21)(7,18)(9,15)(11,22)(12,19)(14,16)(20,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,23)(2,22)(3,24)(4,8)(5,7)(6,9)(10,18)(11,17)(12,16)(13,20)(14,19)(15,21) );

G=PermutationGroup([(1,8),(2,9),(3,7),(4,23),(5,24),(6,22),(10,20),(11,21),(12,19),(13,18),(14,16),(15,17)], [(1,19),(2,10),(3,7),(4,14),(5,17),(6,22),(8,12),(9,20),(11,21),(13,18),(15,24),(16,23)], [(1,12),(2,9),(3,21),(4,16),(5,24),(6,13),(7,11),(8,19),(10,20),(14,23),(15,17),(18,22)], [(1,16),(2,15),(3,7),(4,12),(5,20),(6,22),(8,14),(9,17),(10,24),(11,21),(13,18),(19,23)], [(1,8),(2,17),(3,13),(4,23),(5,10),(6,21),(7,18),(9,15),(11,22),(12,19),(14,16),(20,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,23),(2,22),(3,24),(4,8),(5,7),(6,9),(10,18),(11,17),(12,16),(13,20),(14,19),(15,21)])

G:=TransitiveGroup(24,489);

On 24 points - transitive group 24T490
Generators in S24
(1 15)(2 13)(3 14)(4 24)(5 22)(6 23)(7 16)(8 17)(9 18)(10 21)(11 19)(12 20)
(1 11)(2 12)(4 17)(6 16)(7 23)(8 24)(13 20)(15 19)
(1 11)(3 10)(5 18)(6 16)(7 23)(9 22)(14 21)(15 19)
(1 19)(2 20)(4 8)(6 7)(11 15)(12 13)(16 23)(17 24)
(2 20)(3 21)(4 8)(5 9)(10 14)(12 13)(17 24)(18 22)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 6)(2 5)(3 4)(7 19)(8 21)(9 20)(10 17)(11 16)(12 18)(13 22)(14 24)(15 23)

G:=sub<Sym(24)| (1,15)(2,13)(3,14)(4,24)(5,22)(6,23)(7,16)(8,17)(9,18)(10,21)(11,19)(12,20), (1,11)(2,12)(4,17)(6,16)(7,23)(8,24)(13,20)(15,19), (1,11)(3,10)(5,18)(6,16)(7,23)(9,22)(14,21)(15,19), (1,19)(2,20)(4,8)(6,7)(11,15)(12,13)(16,23)(17,24), (2,20)(3,21)(4,8)(5,9)(10,14)(12,13)(17,24)(18,22), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23)>;

G:=Group( (1,15)(2,13)(3,14)(4,24)(5,22)(6,23)(7,16)(8,17)(9,18)(10,21)(11,19)(12,20), (1,11)(2,12)(4,17)(6,16)(7,23)(8,24)(13,20)(15,19), (1,11)(3,10)(5,18)(6,16)(7,23)(9,22)(14,21)(15,19), (1,19)(2,20)(4,8)(6,7)(11,15)(12,13)(16,23)(17,24), (2,20)(3,21)(4,8)(5,9)(10,14)(12,13)(17,24)(18,22), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23) );

G=PermutationGroup([(1,15),(2,13),(3,14),(4,24),(5,22),(6,23),(7,16),(8,17),(9,18),(10,21),(11,19),(12,20)], [(1,11),(2,12),(4,17),(6,16),(7,23),(8,24),(13,20),(15,19)], [(1,11),(3,10),(5,18),(6,16),(7,23),(9,22),(14,21),(15,19)], [(1,19),(2,20),(4,8),(6,7),(11,15),(12,13),(16,23),(17,24)], [(2,20),(3,21),(4,8),(5,9),(10,14),(12,13),(17,24),(18,22)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,6),(2,5),(3,4),(7,19),(8,21),(9,20),(10,17),(11,16),(12,18),(13,22),(14,24),(15,23)])

G:=TransitiveGroup(24,490);

On 24 points - transitive group 24T491
Generators in S24
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)
(1 15)(2 13)(4 20)(5 21)(8 22)(9 23)(10 18)(11 16)
(1 15)(3 14)(4 20)(6 19)(7 24)(8 22)(10 18)(12 17)
(1 18)(2 16)(4 8)(5 9)(10 15)(11 13)(20 22)(21 23)
(2 16)(3 17)(5 9)(6 7)(11 13)(12 14)(19 24)(21 23)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(2 3)(5 6)(7 9)(11 12)(13 14)(16 17)(19 21)(23 24)

G:=sub<Sym(24)| (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,15)(2,13)(4,20)(5,21)(8,22)(9,23)(10,18)(11,16), (1,15)(3,14)(4,20)(6,19)(7,24)(8,22)(10,18)(12,17), (1,18)(2,16)(4,8)(5,9)(10,15)(11,13)(20,22)(21,23), (2,16)(3,17)(5,9)(6,7)(11,13)(12,14)(19,24)(21,23), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (2,3)(5,6)(7,9)(11,12)(13,14)(16,17)(19,21)(23,24)>;

G:=Group( (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,15)(2,13)(4,20)(5,21)(8,22)(9,23)(10,18)(11,16), (1,15)(3,14)(4,20)(6,19)(7,24)(8,22)(10,18)(12,17), (1,18)(2,16)(4,8)(5,9)(10,15)(11,13)(20,22)(21,23), (2,16)(3,17)(5,9)(6,7)(11,13)(12,14)(19,24)(21,23), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (2,3)(5,6)(7,9)(11,12)(13,14)(16,17)(19,21)(23,24) );

G=PermutationGroup([(1,4),(2,5),(3,6),(7,17),(8,18),(9,16),(10,22),(11,23),(12,24),(13,21),(14,19),(15,20)], [(1,15),(2,13),(4,20),(5,21),(8,22),(9,23),(10,18),(11,16)], [(1,15),(3,14),(4,20),(6,19),(7,24),(8,22),(10,18),(12,17)], [(1,18),(2,16),(4,8),(5,9),(10,15),(11,13),(20,22),(21,23)], [(2,16),(3,17),(5,9),(6,7),(11,13),(12,14),(19,24),(21,23)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(2,3),(5,6),(7,9),(11,12),(13,14),(16,17),(19,21),(23,24)])

G:=TransitiveGroup(24,491);

On 24 points - transitive group 24T492
Generators in S24
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)
(1 15)(2 13)(4 20)(5 21)(8 22)(9 23)(10 18)(11 16)
(1 15)(3 14)(4 20)(6 19)(7 24)(8 22)(10 18)(12 17)
(1 18)(2 16)(4 8)(5 9)(10 15)(11 13)(20 22)(21 23)
(2 16)(3 17)(5 9)(6 7)(11 13)(12 14)(19 24)(21 23)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 15)(2 14)(3 13)(4 20)(5 19)(6 21)(7 23)(8 22)(9 24)(10 18)(11 17)(12 16)

G:=sub<Sym(24)| (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,15)(2,13)(4,20)(5,21)(8,22)(9,23)(10,18)(11,16), (1,15)(3,14)(4,20)(6,19)(7,24)(8,22)(10,18)(12,17), (1,18)(2,16)(4,8)(5,9)(10,15)(11,13)(20,22)(21,23), (2,16)(3,17)(5,9)(6,7)(11,13)(12,14)(19,24)(21,23), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,15)(2,14)(3,13)(4,20)(5,19)(6,21)(7,23)(8,22)(9,24)(10,18)(11,17)(12,16)>;

G:=Group( (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,15)(2,13)(4,20)(5,21)(8,22)(9,23)(10,18)(11,16), (1,15)(3,14)(4,20)(6,19)(7,24)(8,22)(10,18)(12,17), (1,18)(2,16)(4,8)(5,9)(10,15)(11,13)(20,22)(21,23), (2,16)(3,17)(5,9)(6,7)(11,13)(12,14)(19,24)(21,23), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,15)(2,14)(3,13)(4,20)(5,19)(6,21)(7,23)(8,22)(9,24)(10,18)(11,17)(12,16) );

G=PermutationGroup([(1,4),(2,5),(3,6),(7,17),(8,18),(9,16),(10,22),(11,23),(12,24),(13,21),(14,19),(15,20)], [(1,15),(2,13),(4,20),(5,21),(8,22),(9,23),(10,18),(11,16)], [(1,15),(3,14),(4,20),(6,19),(7,24),(8,22),(10,18),(12,17)], [(1,18),(2,16),(4,8),(5,9),(10,15),(11,13),(20,22),(21,23)], [(2,16),(3,17),(5,9),(6,7),(11,13),(12,14),(19,24),(21,23)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,15),(2,14),(3,13),(4,20),(5,19),(6,21),(7,23),(8,22),(9,24),(10,18),(11,17),(12,16)])

G:=TransitiveGroup(24,492);

On 24 points - transitive group 24T493
Generators in S24
(1 4)(2 5)(3 6)(7 22)(8 23)(9 24)(10 17)(11 18)(12 16)(13 21)(14 19)(15 20)
(1 19)(2 10)(3 7)(4 14)(5 17)(6 22)(8 12)(9 20)(11 21)(13 18)(15 24)(16 23)
(1 12)(2 9)(3 21)(4 16)(5 24)(6 13)(7 11)(8 19)(10 20)(14 23)(15 17)(18 22)
(1 16)(2 15)(3 7)(4 12)(5 20)(6 22)(8 14)(9 17)(10 24)(11 21)(13 18)(19 23)
(1 8)(2 17)(3 13)(4 23)(5 10)(6 21)(7 18)(9 15)(11 22)(12 19)(14 16)(20 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(2 3)(5 6)(7 9)(10 21)(11 20)(12 19)(13 17)(14 16)(15 18)(22 24)

G:=sub<Sym(24)| (1,4)(2,5)(3,6)(7,22)(8,23)(9,24)(10,17)(11,18)(12,16)(13,21)(14,19)(15,20), (1,19)(2,10)(3,7)(4,14)(5,17)(6,22)(8,12)(9,20)(11,21)(13,18)(15,24)(16,23), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (1,16)(2,15)(3,7)(4,12)(5,20)(6,22)(8,14)(9,17)(10,24)(11,21)(13,18)(19,23), (1,8)(2,17)(3,13)(4,23)(5,10)(6,21)(7,18)(9,15)(11,22)(12,19)(14,16)(20,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (2,3)(5,6)(7,9)(10,21)(11,20)(12,19)(13,17)(14,16)(15,18)(22,24)>;

G:=Group( (1,4)(2,5)(3,6)(7,22)(8,23)(9,24)(10,17)(11,18)(12,16)(13,21)(14,19)(15,20), (1,19)(2,10)(3,7)(4,14)(5,17)(6,22)(8,12)(9,20)(11,21)(13,18)(15,24)(16,23), (1,12)(2,9)(3,21)(4,16)(5,24)(6,13)(7,11)(8,19)(10,20)(14,23)(15,17)(18,22), (1,16)(2,15)(3,7)(4,12)(5,20)(6,22)(8,14)(9,17)(10,24)(11,21)(13,18)(19,23), (1,8)(2,17)(3,13)(4,23)(5,10)(6,21)(7,18)(9,15)(11,22)(12,19)(14,16)(20,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (2,3)(5,6)(7,9)(10,21)(11,20)(12,19)(13,17)(14,16)(15,18)(22,24) );

G=PermutationGroup([(1,4),(2,5),(3,6),(7,22),(8,23),(9,24),(10,17),(11,18),(12,16),(13,21),(14,19),(15,20)], [(1,19),(2,10),(3,7),(4,14),(5,17),(6,22),(8,12),(9,20),(11,21),(13,18),(15,24),(16,23)], [(1,12),(2,9),(3,21),(4,16),(5,24),(6,13),(7,11),(8,19),(10,20),(14,23),(15,17),(18,22)], [(1,16),(2,15),(3,7),(4,12),(5,20),(6,22),(8,14),(9,17),(10,24),(11,21),(13,18),(19,23)], [(1,8),(2,17),(3,13),(4,23),(5,10),(6,21),(7,18),(9,15),(11,22),(12,19),(14,16),(20,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(2,3),(5,6),(7,9),(10,21),(11,20),(12,19),(13,17),(14,16),(15,18),(22,24)])

G:=TransitiveGroup(24,493);

On 24 points - transitive group 24T508
Generators in S24
(1 11)(2 12)(3 10)(4 17)(5 18)(6 16)(7 23)(8 24)(9 22)(13 20)(14 21)(15 19)
(1 19)(2 12)(3 14)(4 8)(5 22)(6 16)(7 23)(9 18)(10 21)(11 15)(13 20)(17 24)
(1 11)(2 13)(3 21)(4 24)(5 18)(6 7)(8 17)(9 22)(10 14)(12 20)(15 19)(16 23)
(1 15)(2 13)(4 24)(6 23)(7 16)(8 17)(11 19)(12 20)
(2 13)(3 14)(4 24)(5 22)(8 17)(9 18)(10 21)(12 20)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 6)(2 5)(3 4)(7 19)(8 21)(9 20)(10 17)(11 16)(12 18)(13 22)(14 24)(15 23)

G:=sub<Sym(24)| (1,11)(2,12)(3,10)(4,17)(5,18)(6,16)(7,23)(8,24)(9,22)(13,20)(14,21)(15,19), (1,19)(2,12)(3,14)(4,8)(5,22)(6,16)(7,23)(9,18)(10,21)(11,15)(13,20)(17,24), (1,11)(2,13)(3,21)(4,24)(5,18)(6,7)(8,17)(9,22)(10,14)(12,20)(15,19)(16,23), (1,15)(2,13)(4,24)(6,23)(7,16)(8,17)(11,19)(12,20), (2,13)(3,14)(4,24)(5,22)(8,17)(9,18)(10,21)(12,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23)>;

G:=Group( (1,11)(2,12)(3,10)(4,17)(5,18)(6,16)(7,23)(8,24)(9,22)(13,20)(14,21)(15,19), (1,19)(2,12)(3,14)(4,8)(5,22)(6,16)(7,23)(9,18)(10,21)(11,15)(13,20)(17,24), (1,11)(2,13)(3,21)(4,24)(5,18)(6,7)(8,17)(9,22)(10,14)(12,20)(15,19)(16,23), (1,15)(2,13)(4,24)(6,23)(7,16)(8,17)(11,19)(12,20), (2,13)(3,14)(4,24)(5,22)(8,17)(9,18)(10,21)(12,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23) );

G=PermutationGroup([(1,11),(2,12),(3,10),(4,17),(5,18),(6,16),(7,23),(8,24),(9,22),(13,20),(14,21),(15,19)], [(1,19),(2,12),(3,14),(4,8),(5,22),(6,16),(7,23),(9,18),(10,21),(11,15),(13,20),(17,24)], [(1,11),(2,13),(3,21),(4,24),(5,18),(6,7),(8,17),(9,22),(10,14),(12,20),(15,19),(16,23)], [(1,15),(2,13),(4,24),(6,23),(7,16),(8,17),(11,19),(12,20)], [(2,13),(3,14),(4,24),(5,22),(8,17),(9,18),(10,21),(12,20)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,6),(2,5),(3,4),(7,19),(8,21),(9,20),(10,17),(11,16),(12,18),(13,22),(14,24),(15,23)])

G:=TransitiveGroup(24,508);

On 24 points - transitive group 24T509
Generators in S24
(1 15)(2 13)(3 14)(4 24)(5 22)(6 23)(7 16)(8 17)(9 18)(10 21)(11 19)(12 20)
(1 11)(2 12)(4 17)(6 16)(7 23)(8 24)(13 20)(15 19)
(1 11)(3 10)(5 18)(6 16)(7 23)(9 22)(14 21)(15 19)
(1 11)(2 20)(3 14)(4 17)(5 22)(6 7)(8 24)(9 18)(10 21)(12 13)(15 19)(16 23)
(1 15)(2 12)(3 21)(4 8)(5 18)(6 23)(7 16)(9 22)(10 14)(11 19)(13 20)(17 24)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 6)(2 5)(3 4)(7 19)(8 21)(9 20)(10 17)(11 16)(12 18)(13 22)(14 24)(15 23)

G:=sub<Sym(24)| (1,15)(2,13)(3,14)(4,24)(5,22)(6,23)(7,16)(8,17)(9,18)(10,21)(11,19)(12,20), (1,11)(2,12)(4,17)(6,16)(7,23)(8,24)(13,20)(15,19), (1,11)(3,10)(5,18)(6,16)(7,23)(9,22)(14,21)(15,19), (1,11)(2,20)(3,14)(4,17)(5,22)(6,7)(8,24)(9,18)(10,21)(12,13)(15,19)(16,23), (1,15)(2,12)(3,21)(4,8)(5,18)(6,23)(7,16)(9,22)(10,14)(11,19)(13,20)(17,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23)>;

G:=Group( (1,15)(2,13)(3,14)(4,24)(5,22)(6,23)(7,16)(8,17)(9,18)(10,21)(11,19)(12,20), (1,11)(2,12)(4,17)(6,16)(7,23)(8,24)(13,20)(15,19), (1,11)(3,10)(5,18)(6,16)(7,23)(9,22)(14,21)(15,19), (1,11)(2,20)(3,14)(4,17)(5,22)(6,7)(8,24)(9,18)(10,21)(12,13)(15,19)(16,23), (1,15)(2,12)(3,21)(4,8)(5,18)(6,23)(7,16)(9,22)(10,14)(11,19)(13,20)(17,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23) );

G=PermutationGroup([(1,15),(2,13),(3,14),(4,24),(5,22),(6,23),(7,16),(8,17),(9,18),(10,21),(11,19),(12,20)], [(1,11),(2,12),(4,17),(6,16),(7,23),(8,24),(13,20),(15,19)], [(1,11),(3,10),(5,18),(6,16),(7,23),(9,22),(14,21),(15,19)], [(1,11),(2,20),(3,14),(4,17),(5,22),(6,7),(8,24),(9,18),(10,21),(12,13),(15,19),(16,23)], [(1,15),(2,12),(3,21),(4,8),(5,18),(6,23),(7,16),(9,22),(10,14),(11,19),(13,20),(17,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,6),(2,5),(3,4),(7,19),(8,21),(9,20),(10,17),(11,16),(12,18),(13,22),(14,24),(15,23)])

G:=TransitiveGroup(24,509);

On 24 points - transitive group 24T510
Generators in S24
(1 11)(2 12)(3 10)(4 17)(5 18)(6 16)(7 23)(8 24)(9 22)(13 20)(14 21)(15 19)
(1 15)(2 12)(3 21)(4 24)(5 9)(6 16)(7 23)(8 17)(10 14)(11 19)(13 20)(18 22)
(1 11)(2 20)(3 14)(4 8)(5 18)(6 23)(7 16)(9 22)(10 21)(12 13)(15 19)(17 24)
(1 19)(2 13)(3 10)(4 8)(5 18)(6 23)(7 16)(9 22)(11 15)(12 20)(14 21)(17 24)
(1 11)(2 20)(3 14)(4 24)(5 9)(6 16)(7 23)(8 17)(10 21)(12 13)(15 19)(18 22)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 6)(2 5)(3 4)(7 19)(8 21)(9 20)(10 17)(11 16)(12 18)(13 22)(14 24)(15 23)

G:=sub<Sym(24)| (1,11)(2,12)(3,10)(4,17)(5,18)(6,16)(7,23)(8,24)(9,22)(13,20)(14,21)(15,19), (1,15)(2,12)(3,21)(4,24)(5,9)(6,16)(7,23)(8,17)(10,14)(11,19)(13,20)(18,22), (1,11)(2,20)(3,14)(4,8)(5,18)(6,23)(7,16)(9,22)(10,21)(12,13)(15,19)(17,24), (1,19)(2,13)(3,10)(4,8)(5,18)(6,23)(7,16)(9,22)(11,15)(12,20)(14,21)(17,24), (1,11)(2,20)(3,14)(4,24)(5,9)(6,16)(7,23)(8,17)(10,21)(12,13)(15,19)(18,22), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23)>;

G:=Group( (1,11)(2,12)(3,10)(4,17)(5,18)(6,16)(7,23)(8,24)(9,22)(13,20)(14,21)(15,19), (1,15)(2,12)(3,21)(4,24)(5,9)(6,16)(7,23)(8,17)(10,14)(11,19)(13,20)(18,22), (1,11)(2,20)(3,14)(4,8)(5,18)(6,23)(7,16)(9,22)(10,21)(12,13)(15,19)(17,24), (1,19)(2,13)(3,10)(4,8)(5,18)(6,23)(7,16)(9,22)(11,15)(12,20)(14,21)(17,24), (1,11)(2,20)(3,14)(4,24)(5,9)(6,16)(7,23)(8,17)(10,21)(12,13)(15,19)(18,22), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,6)(2,5)(3,4)(7,19)(8,21)(9,20)(10,17)(11,16)(12,18)(13,22)(14,24)(15,23) );

G=PermutationGroup([(1,11),(2,12),(3,10),(4,17),(5,18),(6,16),(7,23),(8,24),(9,22),(13,20),(14,21),(15,19)], [(1,15),(2,12),(3,21),(4,24),(5,9),(6,16),(7,23),(8,17),(10,14),(11,19),(13,20),(18,22)], [(1,11),(2,20),(3,14),(4,8),(5,18),(6,23),(7,16),(9,22),(10,21),(12,13),(15,19),(17,24)], [(1,19),(2,13),(3,10),(4,8),(5,18),(6,23),(7,16),(9,22),(11,15),(12,20),(14,21),(17,24)], [(1,11),(2,20),(3,14),(4,24),(5,9),(6,16),(7,23),(8,17),(10,21),(12,13),(15,19),(18,22)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,6),(2,5),(3,4),(7,19),(8,21),(9,20),(10,17),(11,16),(12,18),(13,22),(14,24),(15,23)])

G:=TransitiveGroup(24,510);

On 24 points - transitive group 24T511
Generators in S24
(1 10)(2 11)(3 12)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(1 7)(3 9)(4 24)(5 22)(10 13)(12 15)(16 20)(18 19)
(2 8)(3 9)(4 24)(6 23)(11 14)(12 15)(17 21)(18 19)
(1 7)(2 8)(4 24)(6 23)(10 13)(11 14)(17 21)(18 19)
(2 8)(3 9)(4 24)(5 22)(11 14)(12 15)(16 20)(18 19)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 17)(2 16)(3 18)(4 15)(5 14)(6 13)(7 21)(8 20)(9 19)(10 23)(11 22)(12 24)

G:=sub<Sym(24)| (1,10)(2,11)(3,12)(4,19)(5,20)(6,21)(7,13)(8,14)(9,15)(16,22)(17,23)(18,24), (1,7)(3,9)(4,24)(5,22)(10,13)(12,15)(16,20)(18,19), (2,8)(3,9)(4,24)(6,23)(11,14)(12,15)(17,21)(18,19), (1,7)(2,8)(4,24)(6,23)(10,13)(11,14)(17,21)(18,19), (2,8)(3,9)(4,24)(5,22)(11,14)(12,15)(16,20)(18,19), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,17)(2,16)(3,18)(4,15)(5,14)(6,13)(7,21)(8,20)(9,19)(10,23)(11,22)(12,24)>;

G:=Group( (1,10)(2,11)(3,12)(4,19)(5,20)(6,21)(7,13)(8,14)(9,15)(16,22)(17,23)(18,24), (1,7)(3,9)(4,24)(5,22)(10,13)(12,15)(16,20)(18,19), (2,8)(3,9)(4,24)(6,23)(11,14)(12,15)(17,21)(18,19), (1,7)(2,8)(4,24)(6,23)(10,13)(11,14)(17,21)(18,19), (2,8)(3,9)(4,24)(5,22)(11,14)(12,15)(16,20)(18,19), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,17)(2,16)(3,18)(4,15)(5,14)(6,13)(7,21)(8,20)(9,19)(10,23)(11,22)(12,24) );

G=PermutationGroup([(1,10),(2,11),(3,12),(4,19),(5,20),(6,21),(7,13),(8,14),(9,15),(16,22),(17,23),(18,24)], [(1,7),(3,9),(4,24),(5,22),(10,13),(12,15),(16,20),(18,19)], [(2,8),(3,9),(4,24),(6,23),(11,14),(12,15),(17,21),(18,19)], [(1,7),(2,8),(4,24),(6,23),(10,13),(11,14),(17,21),(18,19)], [(2,8),(3,9),(4,24),(5,22),(11,14),(12,15),(16,20),(18,19)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,17),(2,16),(3,18),(4,15),(5,14),(6,13),(7,21),(8,20),(9,19),(10,23),(11,22),(12,24)])

G:=TransitiveGroup(24,511);

Polynomial with Galois group C2×C22⋊S4 over ℚ
actionf(x)Disc(f)
12T100x12-6x11+19x10-40x9+63x8-78x7+74x6-51x5+21x4-x3-11x2+9x+1212·54·72·2575
12T101x12-15x10+87x8-245x6+344x4-220x2+49212·72·892·2294
12T103x12-3x10-17x8-25x6-68x4-48x2+64282·32·1016
12T106x12-36x10+519x8-3826x6+15126x4-30096x2+23104250·314·58·196

Matrix representation of C2×C22⋊S4 in GL6(ℤ)

 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1
,
 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0
,
 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 -1 0 0

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,0,0] >;

C2×C22⋊S4 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes S_4
% in TeX

G:=Group("C2xC2^2:S4");
// GroupNames label

G:=SmallGroup(192,1538);
// by ID

G=gap.SmallGroup(192,1538);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,170,185,1264,333,6053,1027,1784]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^3=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,f*c*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=g*b*g=c,c*d=d*c,c*e=e*c,g*c*g=b,f*d*f^-1=g*d*g=d*e=e*d,f*e*f^-1=d,e*g=g*e,g*f*g=f^-1>;
// generators/relations

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