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## G = C2×C42⋊S3order 192 = 26·3

### Direct product of C2 and C42⋊S3

Aliases: C2×C42⋊S3, C422D6, C23.10S4, (C2×C42)⋊S3, C42⋊C34C22, C22.1(C2×S4), (C2×C42⋊C3)⋊3C2, SmallGroup(192,944)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C42⋊C3 — C2×C42⋊S3
 Chief series C1 — C22 — C42 — C42⋊C3 — C42⋊S3 — C2×C42⋊S3
 Lower central C42⋊C3 — C2×C42⋊S3
 Upper central C1 — C2

Generators and relations for C2×C42⋊S3
G = < a,b,c,d,e | a2=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=c, dcd-1=b-1c-1, ece=b, ede=d-1 >

Subgroups: 408 in 84 conjugacy classes, 11 normal (9 characteristic)
C1, C2, C2 [×4], C3, C4 [×6], C22, C22 [×6], S3 [×2], C6, C8 [×2], C2×C4 [×11], D4 [×7], Q8 [×3], C23, C23, A4, D6, C42, C42, C2×C8, M4(2) [×3], C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], S4 [×2], C2×A4, C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C42⋊C3, C2×S4, C2×C4≀C2, C42⋊S3 [×2], C2×C42⋊C3, C2×C42⋊S3
Quotients: C1, C2 [×3], C22, S3, D6, S4, C2×S4, C42⋊S3, C2×C42⋊S3

Character table of C2×C42⋊S3

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6 8A 8B 8C 8D size 1 1 3 3 12 12 32 3 3 3 3 6 6 12 12 32 12 12 12 12 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 0 0 -1 2 2 2 2 2 2 0 0 -1 0 0 0 0 orthogonal lifted from S3 ρ6 2 -2 2 -2 0 0 -1 -2 2 2 -2 -2 2 0 0 1 0 0 0 0 orthogonal lifted from D6 ρ7 3 3 3 3 1 1 0 -1 -1 -1 -1 -1 -1 1 1 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ8 3 -3 3 -3 1 -1 0 1 -1 -1 1 1 -1 -1 1 0 -1 1 -1 1 orthogonal lifted from C2×S4 ρ9 3 -3 3 -3 -1 1 0 1 -1 -1 1 1 -1 1 -1 0 1 -1 1 -1 orthogonal lifted from C2×S4 ρ10 3 3 3 3 -1 -1 0 -1 -1 -1 -1 -1 -1 -1 -1 0 1 1 1 1 orthogonal lifted from S4 ρ11 3 3 -1 -1 -1 -1 0 -1+2i -1+2i -1-2i -1-2i 1 1 1 1 0 -i -i i i complex lifted from C42⋊S3 ρ12 3 -3 -1 1 1 -1 0 1+2i -1-2i -1+2i 1-2i -1 1 1 -1 0 -i i i -i complex faithful ρ13 3 3 -1 -1 -1 -1 0 -1-2i -1-2i -1+2i -1+2i 1 1 1 1 0 i i -i -i complex lifted from C42⋊S3 ρ14 3 -3 -1 1 -1 1 0 1-2i -1+2i -1-2i 1+2i -1 1 -1 1 0 -i i i -i complex faithful ρ15 3 3 -1 -1 1 1 0 -1-2i -1-2i -1+2i -1+2i 1 1 -1 -1 0 -i -i i i complex lifted from C42⋊S3 ρ16 3 3 -1 -1 1 1 0 -1+2i -1+2i -1-2i -1-2i 1 1 -1 -1 0 i i -i -i complex lifted from C42⋊S3 ρ17 3 -3 -1 1 -1 1 0 1+2i -1-2i -1+2i 1-2i -1 1 -1 1 0 i -i -i i complex faithful ρ18 3 -3 -1 1 1 -1 0 1-2i -1+2i -1-2i 1+2i -1 1 1 -1 0 i -i -i i complex faithful ρ19 6 6 -2 -2 0 0 0 2 2 2 2 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from C42⋊S3 ρ20 6 -6 -2 2 0 0 0 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C2×C42⋊S3
On 12 points - transitive group 12T95
Generators in S12
(1 2)(3 4)(5 7)(6 8)(9 11)(10 12)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 4 2 3)(5 7)(6 8)(9 10 11 12)
(1 12 8)(2 10 6)(3 11 7)(4 9 5)
(1 8)(2 6)(3 7)(4 5)

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

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

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

G:=TransitiveGroup(12,95);

On 12 points - transitive group 12T96
Generators in S12
(1 4)(2 3)(5 7)(6 8)(9 11)(10 12)
(5 6 7 8)(9 10 11 12)
(1 3 4 2)(9 12 11 10)
(1 12 7)(2 11 8)(3 9 6)(4 10 5)
(1 7)(2 6)(3 8)(4 5)(9 11)

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

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

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

G:=TransitiveGroup(12,96);

On 12 points - transitive group 12T97
Generators in S12
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)
(5 6 7 8)(9 10 11 12)
(1 2 3 4)(9 12 11 10)
(1 12 6)(2 9 5)(3 10 8)(4 11 7)
(1 7)(2 8)(3 5)(4 6)(9 10)(11 12)

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

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

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

G:=TransitiveGroup(12,97);

On 24 points - transitive group 24T471
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 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 7 2 8)(3 6 4 5)(9 10 11 12)(13 14 15 16)(17 19)(18 20)(21 23)(22 24)
(1 9 18)(2 11 20)(3 16 24)(4 14 22)(5 15 23)(6 13 21)(7 10 19)(8 12 17)
(1 22)(2 24)(3 20)(4 18)(5 19)(6 17)(7 23)(8 21)(9 14)(10 15)(11 16)(12 13)

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

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

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

G:=TransitiveGroup(24,471);

On 24 points - transitive group 24T472
Generators in S24
(1 4)(2 3)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 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 8 4 6)(2 7 3 5)(9 13 11 15)(10 14 12 16)(17 24)(18 21)(19 22)(20 23)
(1 9 21)(2 14 18)(3 16 20)(4 11 23)(5 13 22)(6 12 19)(7 15 24)(8 10 17)
(1 21)(2 18)(3 20)(4 23)(5 17)(6 24)(7 19)(8 22)(10 13)(12 15)

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

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

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

G:=TransitiveGroup(24,472);

On 24 points - transitive group 24T473
Generators in S24
(1 8)(2 7)(3 5)(4 6)(9 15)(10 16)(11 13)(12 14)(17 24)(18 21)(19 22)(20 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 4 2 3)(5 8 6 7)(9 10 11 12)(13 14 15 16)(17 19)(18 20)(21 23)(22 24)
(1 16 24)(2 14 22)(3 15 23)(4 13 21)(5 9 20)(6 11 18)(7 12 19)(8 10 17)
(1 24)(2 22)(3 23)(4 21)(5 20)(6 18)(7 19)(8 17)

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

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

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

G:=TransitiveGroup(24,473);

On 24 points - transitive group 24T474
Generators in S24
(1 7)(2 8)(3 6)(4 5)(9 24)(10 21)(11 22)(12 23)(13 18)(14 19)(15 20)(16 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 3)(5 8 6 7)(9 11)(10 12)(13 14 15 16)(17 18 19 20)(21 23)(22 24)
(1 18 22)(2 20 24)(3 17 21)(4 19 23)(5 14 12)(6 16 10)(7 13 11)(8 15 9)
(1 24)(2 22)(3 23)(4 21)(5 10)(6 12)(7 9)(8 11)(13 15)(14 16)(17 19)(18 20)

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

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

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

G:=TransitiveGroup(24,474);

On 24 points - transitive group 24T475
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 23)(10 24)(11 21)(12 22)(13 19)(14 20)(15 17)(16 18)
(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 7 3 5)(2 8 4 6)(9 22 11 24)(10 23 12 21)(13 19)(14 20)(15 17)(16 18)
(1 9 17)(2 23 15)(3 11 19)(4 21 13)(5 12 16)(6 22 18)(7 10 14)(8 24 20)
(1 13)(2 19)(3 15)(4 17)(5 16)(6 18)(7 14)(8 20)(9 21)(11 23)

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

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

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

G:=TransitiveGroup(24,475);

On 24 points - transitive group 24T476
Generators in S24
(1 6)(2 4)(3 8)(5 7)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 2 6 4)(3 7 8 5)(9 12 11 10)(17 20 19 18)
(1 18 22)(2 19 21)(3 12 16)(4 17 23)(5 11 13)(6 20 24)(7 9 15)(8 10 14)
(1 14)(2 15)(3 24)(4 13)(5 23)(6 16)(7 21)(8 22)(9 19)(10 18)(11 17)(12 20)

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

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

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

G:=TransitiveGroup(24,476);

On 24 points - transitive group 24T477
Generators in S24
(1 8)(2 7)(3 6)(4 5)(9 22)(10 23)(11 24)(12 21)(13 17)(14 18)(15 19)(16 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 3 7 5)(2 4 8 6)(9 23 11 21)(10 24 12 22)(13 19)(14 20)(15 17)(16 18)
(1 11 16)(2 22 18)(3 12 17)(4 23 15)(5 10 19)(6 21 13)(7 9 14)(8 24 20)
(1 16)(2 18)(3 13)(4 19)(5 15)(6 17)(7 14)(8 20)(10 23)(12 21)

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

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

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

G:=TransitiveGroup(24,477);

On 24 points - transitive group 24T478
Generators in S24
(1 3)(2 4)(5 7)(6 8)(9 13)(10 14)(11 15)(12 16)(17 22)(18 23)(19 24)(20 21)
(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 4 7)(2 5 3 8)(9 14 11 16)(10 15 12 13)(17 24)(18 21)(19 22)(20 23)
(1 12 19)(2 14 22)(3 16 24)(4 10 17)(5 15 18)(6 9 21)(7 11 23)(8 13 20)
(1 17)(2 24)(3 22)(4 19)(5 21)(6 18)(7 20)(8 23)(9 15)(10 12)(11 13)(14 16)

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

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

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

G:=TransitiveGroup(24,478);

On 24 points - transitive group 24T479
Generators in S24
(1 5)(2 8)(3 7)(4 6)(9 22)(10 23)(11 24)(12 21)(13 18)(14 19)(15 20)(16 17)
(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 2 4 3)(5 8 6 7)(13 16 15 14)(17 20 19 18)
(1 19 21)(2 20 24)(3 18 22)(4 17 23)(5 14 12)(6 16 10)(7 13 9)(8 15 11)
(1 12)(2 9)(3 11)(4 10)(5 21)(6 23)(7 24)(8 22)(13 20)(14 19)(15 18)(16 17)

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

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

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

G:=TransitiveGroup(24,479);

On 24 points - transitive group 24T480
Generators in S24
(1 8)(2 6)(3 5)(4 7)(9 19)(10 20)(11 17)(12 18)(13 21)(14 22)(15 23)(16 24)
(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 2 4 3)(5 8 6 7)(9 12 11 10)(17 20 19 18)
(1 20 23)(2 17 22)(3 19 24)(4 18 21)(5 9 16)(6 11 14)(7 12 13)(8 10 15)
(1 24)(2 21)(3 23)(4 22)(5 15)(6 13)(7 14)(8 16)(9 10)(11 12)(17 18)(19 20)

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

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

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

G:=TransitiveGroup(24,480);

Polynomial with Galois group C2×C42⋊S3 over ℚ
actionf(x)Disc(f)
12T95x12-20x10+107x8-196x6+112x4-22x2+1224·312·134·174·414
12T96x12-3x4-4-238·316
12T97x12-32x10+391x8-2248x6+5951x4-5622x2+81224·38·58·536·15594

Matrix representation of C2×C42⋊S3 in GL3(𝔽5) generated by

 4 0 0 0 4 0 0 0 4
,
 3 0 0 0 4 0 0 0 3
,
 3 0 0 0 3 0 0 0 4
,
 0 4 0 0 0 1 4 0 0
,
 4 0 0 0 0 4 0 4 0
G:=sub<GL(3,GF(5))| [4,0,0,0,4,0,0,0,4],[3,0,0,0,4,0,0,0,3],[3,0,0,0,3,0,0,0,4],[0,0,4,4,0,0,0,1,0],[4,0,0,0,0,4,0,4,0] >;

C2×C42⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes S_3
% in TeX

G:=Group("C2xC4^2:S3");
// GroupNames label

G:=SmallGroup(192,944);
// by ID

G=gap.SmallGroup(192,944);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,170,675,185,360,424,1173,102,6053,1027,1784]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=c,d*c*d^-1=b^-1*c^-1,e*c*e=b,e*d*e=d^-1>;
// generators/relations

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