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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, C3, C4, C22, C22, C6, C2×C4, D4, C23, C23, C23, A4, C2×C6, C42, C42, C22×C4, C2×D4, C24, C24, C2×A4, C2×C42, C41D4, C41D4, C22×D4, C42⋊C3, C22×A4, C2×C41D4, C2×C42⋊C3, C23.A4, C2×C23.A4
Quotients: C1, C2, C3, C22, C6, A4, C2×C6, C2×A4, 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 3)(2 4)(5 8)(6 7)(9 10)(11 12)
(1 2)(3 4)(5 7)(6 8)
(1 2)(3 4)(9 11)(10 12)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 4 2 3)(5 8 7 6)
(1 12 5)(2 10 7)(3 11 8)(4 9 6)

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

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

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

G:=TransitiveGroup(12,89);

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

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

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

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

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

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

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

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

G:=TransitiveGroup(24,453);

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

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

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

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

G:=TransitiveGroup(24,454);

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

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

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

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

G:=TransitiveGroup(24,455);

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

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

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

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

G:=TransitiveGroup(24,456);

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

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

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

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

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

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

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

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

G:=TransitiveGroup(24,464);

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

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

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

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

G:=TransitiveGroup(24,465);

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

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

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

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

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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