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## G = C22×S4order 96 = 25·3

### Direct product of C22 and S4

Aliases: C22×S4, C23⋊D6, A4⋊C23, C242S3, (C2×A4)⋊C22, C22⋊(C22×S3), (C22×A4)⋊3C2, SmallGroup(96,226)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×S4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 420 in 131 conjugacy classes, 26 normal (7 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C23, A4, D6, C2×C6, C22×C4, C2×D4, C24, C24, S4, C2×A4, C22×S3, C22×D4, C2×S4, C22×A4, C22×S4
Quotients: C1, C2, C22, S3, C23, D6, S4, C22×S3, C2×S4, C22×S4

Character table of C22×S4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 6A 6B 6C size 1 1 1 1 3 3 3 3 6 6 6 6 8 6 6 6 6 8 8 8 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 -1 -1 -1 -1 1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ6 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ7 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 -1 -1 1 linear of order 2 ρ8 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ9 2 -2 2 -2 -2 2 -2 2 0 0 0 0 -1 0 0 0 0 1 1 -1 orthogonal lifted from D6 ρ10 2 2 -2 -2 -2 -2 2 2 0 0 0 0 -1 0 0 0 0 1 -1 1 orthogonal lifted from D6 ρ11 2 2 2 2 2 2 2 2 0 0 0 0 -1 0 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ12 2 -2 -2 2 2 -2 -2 2 0 0 0 0 -1 0 0 0 0 -1 1 1 orthogonal lifted from D6 ρ13 3 -3 3 -3 1 -1 1 -1 1 -1 1 -1 0 -1 1 1 -1 0 0 0 orthogonal lifted from C2×S4 ρ14 3 -3 -3 3 -1 1 1 -1 1 1 -1 -1 0 -1 -1 1 1 0 0 0 orthogonal lifted from C2×S4 ρ15 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 0 1 1 1 1 0 0 0 orthogonal lifted from S4 ρ16 3 3 -3 -3 1 1 -1 -1 -1 1 1 -1 0 1 -1 1 -1 0 0 0 orthogonal lifted from C2×S4 ρ17 3 3 3 3 -1 -1 -1 -1 1 1 1 1 0 -1 -1 -1 -1 0 0 0 orthogonal lifted from S4 ρ18 3 3 -3 -3 1 1 -1 -1 1 -1 -1 1 0 -1 1 -1 1 0 0 0 orthogonal lifted from C2×S4 ρ19 3 -3 3 -3 1 -1 1 -1 -1 1 -1 1 0 1 -1 -1 1 0 0 0 orthogonal lifted from C2×S4 ρ20 3 -3 -3 3 -1 1 1 -1 -1 -1 1 1 0 1 1 -1 -1 0 0 0 orthogonal lifted from C2×S4

Permutation representations of C22×S4
On 12 points - transitive group 12T48
Generators in S12
(1 9)(2 7)(3 8)(4 12)(5 10)(6 11)
(1 11)(2 12)(3 10)(4 7)(5 8)(6 9)
(1 9)(2 7)(4 12)(6 11)
(2 7)(3 8)(4 12)(5 10)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 9)(2 8)(3 7)(4 10)(5 12)(6 11)

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

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

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

G:=TransitiveGroup(12,48);

On 16 points - transitive group 16T182
Generators in S16
(1 2)(3 4)(5 9)(6 10)(7 8)(11 14)(12 15)(13 16)
(1 3)(2 4)(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)
(1 5)(2 9)(3 14)(4 11)(6 7)(8 10)(12 13)(15 16)
(1 6)(2 10)(3 15)(4 12)(5 7)(8 9)(11 13)(14 16)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
(1 3)(2 4)(5 16)(6 15)(7 14)(8 11)(9 13)(10 12)

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

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

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

G:=TransitiveGroup(16,182);

On 24 points - transitive group 24T125
Generators in S24
(1 14)(2 15)(3 13)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 16)(11 17)(12 18)
(1 11)(2 12)(3 10)(4 22)(5 23)(6 24)(7 19)(8 20)(9 21)(13 16)(14 17)(15 18)
(1 11)(2 15)(3 16)(4 20)(5 23)(6 7)(8 22)(9 21)(10 13)(12 18)(14 17)(19 24)
(1 17)(2 12)(3 13)(4 8)(5 21)(6 24)(7 19)(9 23)(10 16)(11 14)(15 18)(20 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 20)(2 19)(3 21)(4 14)(5 13)(6 15)(7 12)(8 11)(9 10)(16 23)(17 22)(18 24)

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

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

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

G:=TransitiveGroup(24,125);

On 24 points - transitive group 24T126
Generators in S24
(1 21)(2 19)(3 20)(4 8)(5 9)(6 7)(10 13)(11 14)(12 15)(16 23)(17 24)(18 22)
(1 7)(2 8)(3 9)(4 19)(5 20)(6 21)(10 22)(11 23)(12 24)(13 18)(14 16)(15 17)
(1 14)(2 12)(3 20)(4 17)(5 9)(6 23)(7 16)(8 24)(10 13)(11 21)(15 19)(18 22)
(1 21)(2 15)(3 10)(4 24)(5 18)(6 7)(8 17)(9 22)(11 14)(12 19)(13 20)(16 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)(4 5)(8 9)(10 15)(11 14)(12 13)(16 23)(17 22)(18 24)(19 20)

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

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

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

G:=TransitiveGroup(24,126);

On 24 points - transitive group 24T150
Generators in S24
(1 9)(2 7)(3 8)(4 22)(5 23)(6 24)(10 14)(11 15)(12 13)(16 19)(17 20)(18 21)
(1 11)(2 12)(3 10)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(2 7)(3 8)(4 22)(6 24)(10 14)(12 13)(16 19)(18 21)
(1 9)(3 8)(4 22)(5 23)(10 14)(11 15)(16 19)(17 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 22)(2 24)(3 23)(4 9)(5 8)(6 7)(10 17)(11 16)(12 18)(13 21)(14 20)(15 19)

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

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

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

G:=TransitiveGroup(24,150);

On 24 points - transitive group 24T151
Generators in S24
(1 17)(2 18)(3 16)(4 13)(5 14)(6 15)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 11)(2 12)(3 10)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(1 9)(2 7)(4 24)(6 23)(11 15)(12 13)(17 21)(18 19)
(2 7)(3 8)(4 24)(5 22)(10 14)(12 13)(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 11)(2 10)(3 12)(4 20)(5 19)(6 21)(7 14)(8 13)(9 15)(16 24)(17 23)(18 22)

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

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

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

G:=TransitiveGroup(24,151);

On 24 points - transitive group 24T152
Generators in S24
(1 17)(2 18)(3 16)(4 13)(5 14)(6 15)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 11)(2 12)(3 10)(4 19)(5 20)(6 21)(7 13)(8 14)(9 15)(16 22)(17 23)(18 24)
(1 9)(2 7)(4 24)(6 23)(11 15)(12 13)(17 21)(18 19)
(2 7)(3 8)(4 24)(5 22)(10 14)(12 13)(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 15)(2 14)(3 13)(4 16)(5 18)(6 17)(7 10)(8 12)(9 11)(19 22)(20 24)(21 23)

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

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

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

G:=TransitiveGroup(24,152);

C22×S4 is a maximal subgroup of
C24.5D6
C22×S4 is a maximal quotient of
C24.10D6  D42S4  Q84S4  GL2(𝔽3)⋊C22  Q8.6S4  Q8.7S4  D4.4S4  D4.5S4

Polynomial with Galois group C22×S4 over ℚ
actionf(x)Disc(f)
12T48x12-15x10+85x8-224x6+270x4-120x2+4230·56·474

Matrix representation of C22×S4 in GL5(ℤ)

 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 -1 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 -1 0 0 0 1 0 -1
,
 0 -1 0 0 0 1 -1 0 0 0 0 0 1 0 -2 0 0 0 0 -1 0 0 0 1 -1
,
 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,Integers())| [1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,-1,0,-1,0,0,0,-1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,-1,0,0,0,0,0,-1],[0,1,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,-2,-1,-1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C22×S4 in GAP, Magma, Sage, TeX

C_2^2\times S_4
% in TeX

G:=Group("C2^2xS4");
// GroupNames label

G:=SmallGroup(96,226);
// by ID

G=gap.SmallGroup(96,226);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,2,387,1444,202,869,347]);
// Polycyclic

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

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