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G = C2×C23.A4order 192 = 26·3

Direct product of C2 and C23.A4

Aliases: C2×C23.A4, C24.5A4, C41D42C6, (C2×C42)⋊2C6, C423(C2×C6), C42⋊C35C22, C23.4(C2×A4), C22.4(C22×A4), (C2×C41D4)⋊C3, (C2×C42⋊C3)⋊2C2, SmallGroup(192,1002)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C2×C23.A4
 Chief series C1 — C22 — C42 — C42⋊C3 — C23.A4 — C2×C23.A4
 Lower central C42 — C2×C23.A4
 Upper central C1 — C2

Generators and relations for C2×C23.A4
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g3=1, e2=dc=gcg-1=cd, f2=gdg-1=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fbf-1=bc=cb, bd=db, ebe-1=bcd, bg=gb, ce=ec, cf=fc, gfg-1=de=ed, df=fd, ef=fe, geg-1=cef >

Subgroups: 546 in 108 conjugacy classes, 17 normal (11 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22, C22 [×16], C6 [×3], C2×C4 [×6], D4 [×16], C23, C23 [×2], C23 [×10], A4, C2×C6, C42, C42, C22×C4, C2×D4 [×16], C24, C24, C2×A4 [×3], C2×C42, C41D4 [×2], C41D4 [×2], C22×D4 [×2], C42⋊C3, C22×A4, C2×C41D4, C2×C42⋊C3, C23.A4 [×2], C2×C23.A4
Quotients: C1, C2 [×3], C3, C22, C6 [×3], A4, C2×C6, C2×A4 [×3], C22×A4, C23.A4, C2×C23.A4

Character table of C2×C23.A4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F size 1 1 3 3 4 4 12 12 16 16 6 6 6 6 16 16 16 16 16 16 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 1 1 1 1 -1 -1 -1 -1 ζ3 ζ32 1 1 1 1 ζ6 ζ6 ζ3 ζ32 ζ65 ζ65 linear of order 6 ρ6 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 linear of order 3 ρ7 1 -1 -1 1 1 -1 1 -1 ζ3 ζ32 1 1 -1 -1 ζ6 ζ32 ζ65 ζ6 ζ65 ζ3 linear of order 6 ρ8 1 1 1 1 -1 -1 -1 -1 ζ32 ζ3 1 1 1 1 ζ65 ζ65 ζ32 ζ3 ζ6 ζ6 linear of order 6 ρ9 1 -1 -1 1 1 -1 1 -1 ζ32 ζ3 1 1 -1 -1 ζ65 ζ3 ζ6 ζ65 ζ6 ζ32 linear of order 6 ρ10 1 -1 -1 1 -1 1 -1 1 ζ3 ζ32 1 1 -1 -1 ζ32 ζ6 ζ65 ζ6 ζ3 ζ65 linear of order 6 ρ11 1 -1 -1 1 -1 1 -1 1 ζ32 ζ3 1 1 -1 -1 ζ3 ζ65 ζ6 ζ65 ζ32 ζ6 linear of order 6 ρ12 1 1 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 linear of order 3 ρ13 3 -3 -3 3 -3 3 1 -1 0 0 -1 -1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 3 3 3 3 3 -1 -1 0 0 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ15 3 -3 -3 3 3 -3 -1 1 0 0 -1 -1 1 1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ16 3 3 3 3 -3 -3 1 1 0 0 -1 -1 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ17 6 6 -2 -2 0 0 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from C23.A4 ρ18 6 6 -2 -2 0 0 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 orthogonal lifted from C23.A4 ρ19 6 -6 2 -2 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal faithful ρ20 6 -6 2 -2 0 0 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 orthogonal faithful

Permutation representations of C2×C23.A4
On 12 points - transitive group 12T89
Generators in S12
(1 2)(3 4)(5 7)(6 8)(9 11)(10 12)
(1 4)(2 3)(5 8)(6 7)(9 10)(11 12)
(1 2)(3 4)(9 11)(10 12)
(1 2)(3 4)(5 7)(6 8)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 4 2 3)(9 12 11 10)
(1 8 9)(2 6 11)(3 7 12)(4 5 10)

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

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

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

G:=TransitiveGroup(12,89);

On 12 points - transitive group 12T92
Generators in S12
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)
(2 4)(6 8)(9 11)
(1 3)(2 4)(9 11)(10 12)
(1 3)(2 4)(5 7)(6 8)
(5 6 7 8)(9 10 11 12)
(1 2 3 4)(5 7)(6 8)(9 10 11 12)
(1 5 12)(2 8 9)(3 7 10)(4 6 11)

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

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

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

G:=TransitiveGroup(12,92);

On 24 points - transitive group 24T453
Generators in S24
(1 3)(2 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 7)(2 6)(3 8)(4 5)(9 21)(10 24)(11 23)(12 22)(13 17)(14 20)(15 19)(16 18)
(1 3)(2 4)(5 6)(7 8)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(5 6)(7 8)(9 11)(10 12)(21 23)(22 24)
(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4 3 2)(5 7 6 8)(9 11)(10 12)(13 14 15 16)(17 18 19 20)(21 23)(22 24)
(1 22 14)(2 23 13)(3 24 16)(4 21 15)(5 9 19)(6 11 17)(7 12 20)(8 10 18)

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

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

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

G:=TransitiveGroup(24,453);

On 24 points - transitive group 24T454
Generators in S24
(1 8)(2 7)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 5)(2 6)(3 7)(4 8)(9 23)(10 22)(11 21)(12 24)(13 16)(14 15)(17 20)(18 19)
(1 8)(2 7)(3 6)(4 5)(13 15)(14 16)(17 19)(18 20)
(1 8)(2 7)(3 6)(4 5)(9 11)(10 12)(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 5 8 4)(2 6 7 3)(9 22)(10 23)(11 24)(12 21)(13 20 15 18)(14 17 16 19)
(1 11 18)(2 22 16)(3 12 15)(4 23 17)(5 21 19)(6 10 13)(7 24 14)(8 9 20)

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

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

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

G:=TransitiveGroup(24,454);

On 24 points - transitive group 24T455
Generators in S24
(1 8)(2 4)(3 7)(5 6)(9 24)(10 21)(11 22)(12 23)(13 17)(14 18)(15 19)(16 20)
(1 4)(2 8)(3 5)(6 7)(9 23)(10 22)(11 21)(12 24)(13 18)(14 17)(15 20)(16 19)
(1 3)(2 6)(4 5)(7 8)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 6)(4 5)(7 8)(9 11)(10 12)(21 23)(22 24)
(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 6 3 2)(4 8 5 7)(9 11)(10 12)(13 14 15 16)(17 18 19 20)(21 23)(22 24)
(1 21 20)(2 22 19)(3 23 18)(4 11 15)(5 9 13)(6 24 17)(7 12 14)(8 10 16)

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

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

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

G:=TransitiveGroup(24,455);

On 24 points - transitive group 24T456
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(1 3)(2 4)(5 7)(6 8)(9 15)(10 14)(11 13)(12 16)(17 18)(19 20)(21 24)(22 23)
(1 8)(2 7)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)
(1 8)(2 7)(3 6)(4 5)(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 5 8 4)(2 6 7 3)(9 15 11 13)(10 16 12 14)(17 24)(18 21)(19 22)(20 23)
(1 24 13)(2 19 12)(3 21 11)(4 20 16)(5 18 14)(6 23 9)(7 17 10)(8 22 15)

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

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

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

G:=TransitiveGroup(24,456);

On 24 points - transitive group 24T463
Generators in S24
(1 8)(2 7)(3 5)(4 6)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(3 5)(4 6)(10 12)(14 16)(18 20)(22 24)
(1 8)(2 7)(3 5)(4 6)(13 15)(14 16)(17 19)(18 20)
(1 8)(2 7)(3 5)(4 6)(9 11)(10 12)(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 6 8 4)(2 5 7 3)(9 23)(10 24)(11 21)(12 22)(13 20 15 18)(14 17 16 19)
(1 11 17)(2 23 15)(3 12 14)(4 24 20)(5 10 16)(6 22 18)(7 21 13)(8 9 19)

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

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

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

G:=TransitiveGroup(24,463);

On 24 points - transitive group 24T464
Generators in S24
(1 8)(2 7)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 7)(2 8)(3 4)(5 6)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)
(1 8)(2 7)(3 6)(4 5)(9 11)(10 12)(17 19)(18 20)
(1 8)(2 7)(3 6)(4 5)(13 15)(14 16)(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 6 8 3)(2 5 7 4)(9 17 11 19)(10 18 12 20)(13 23)(14 24)(15 21)(16 22)
(1 14 12)(2 22 19)(3 23 11)(4 15 18)(5 13 20)(6 21 9)(7 24 17)(8 16 10)

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

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

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

G:=TransitiveGroup(24,464);

On 24 points - transitive group 24T465
Generators in S24
(1 7)(2 8)(3 5)(4 6)(9 16)(10 13)(11 14)(12 15)(17 24)(18 21)(19 22)(20 23)
(1 7)(2 8)(3 4)(5 6)(9 14)(10 13)(11 16)(12 15)(17 24)(18 23)(19 22)(20 21)
(1 8)(2 7)(3 6)(4 5)(9 11)(10 12)(13 15)(14 16)
(1 8)(2 7)(3 6)(4 5)(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 5 8 4)(2 6 7 3)(9 15 11 13)(10 16 12 14)(17 24)(18 21)(19 22)(20 23)
(1 24 13)(2 19 12)(3 21 11)(4 20 16)(5 18 14)(6 23 9)(7 17 10)(8 22 15)

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

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

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

G:=TransitiveGroup(24,465);

On 24 points - transitive group 24T466
Generators in S24
(1 7)(2 8)(3 6)(4 5)(9 24)(10 21)(11 22)(12 23)(13 19)(14 20)(15 17)(16 18)
(1 2)(7 8)(9 11)(13 15)(17 19)(22 24)
(1 2)(3 4)(5 6)(7 8)(13 15)(14 16)(17 19)(18 20)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(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 4 2 3)(5 8 6 7)(13 16 15 14)(17 20 19 18)
(1 22 15)(2 24 13)(3 21 14)(4 23 16)(5 12 18)(6 10 20)(7 11 17)(8 9 19)

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

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

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

G:=TransitiveGroup(24,466);

Polynomial with Galois group C2×C23.A4 over ℚ
actionf(x)Disc(f)
12T89x12-x8-2x4+1224·78
12T92x12-12x8+35x4-25-224·514·138

Matrix representation of C2×C23.A4 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
,
 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
,
 -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
,
 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 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

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],[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],[-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],[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],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0] >;

C2×C23.A4 in GAP, Magma, Sage, TeX

C_2\times C_2^3.A_4
% in TeX

G:=Group("C2xC2^3.A4");
// GroupNames label

G:=SmallGroup(192,1002);
// by ID

G=gap.SmallGroup(192,1002);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,4371,185,360,2524,1173,102,1027,1784]);
// Polycyclic

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

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