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## G = C22×C3⋊S4order 288 = 25·32

### Direct product of C22 and C3⋊S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — C22×C3⋊S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C3⋊S4 — C2×C3⋊S4 — C22×C3⋊S4
 Lower central C3×A4 — C22×C3⋊S4
 Upper central C1 — C22

Generators and relations for C22×C3⋊S4
G = < a,b,c,d,e,f,g | a2=b2=c3=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg=c-1, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 1876 in 326 conjugacy classes, 51 normal (11 characteristic)
C1, C2 [×3], C2 [×8], C3, C3 [×3], C4 [×4], C22 [×2], C22 [×26], S3 [×16], C6 [×3], C6 [×13], C2×C4 [×6], D4 [×16], C23 [×3], C23 [×14], C32, Dic3 [×4], A4 [×3], D6 [×34], C2×C6 [×2], C2×C6 [×13], C22×C4, C2×D4 [×12], C24, C24, C3⋊S3 [×4], C3×C6 [×3], C2×Dic3 [×6], C3⋊D4 [×16], S4 [×12], C2×A4 [×9], C22×S3 [×13], C22×C6 [×3], C22×C6 [×4], C22×D4, C3×A4, C2×C3⋊S3 [×6], C62, C22×Dic3, C2×C3⋊D4 [×12], C2×S4 [×18], C22×A4 [×3], S3×C23, C23×C6, C3⋊S4 [×4], C6×A4 [×3], C22×C3⋊S3, C22×C3⋊D4, C22×S4 [×3], C2×C3⋊S4 [×6], A4×C2×C6, C22×C3⋊S4
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], C23, D6 [×12], C3⋊S3, S4, C22×S3 [×4], C2×C3⋊S3 [×3], C2×S4 [×3], C3⋊S4, C22×C3⋊S3, C22×S4, C2×C3⋊S4 [×3], C22×C3⋊S4

Smallest permutation representation of C22×C3⋊S4
On 36 points
Generators in S36
(1 25)(2 26)(3 27)(4 16)(5 17)(6 18)(7 22)(8 23)(9 24)(10 31)(11 32)(12 33)(13 28)(14 29)(15 30)(19 34)(20 35)(21 36)
(1 22)(2 23)(3 24)(4 19)(5 20)(6 21)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)(28 29 30)(31 32 33)(34 35 36)
(1 7)(2 8)(3 9)(10 13)(11 14)(12 15)(22 25)(23 26)(24 27)(28 31)(29 32)(30 33)
(4 34)(5 35)(6 36)(10 13)(11 14)(12 15)(16 19)(17 20)(18 21)(28 31)(29 32)(30 33)
(1 10 16)(2 11 17)(3 12 18)(4 25 31)(5 26 32)(6 27 33)(7 13 19)(8 14 20)(9 15 21)(22 28 34)(23 29 35)(24 30 36)
(1 7)(2 9)(3 8)(4 28)(5 30)(6 29)(10 19)(11 21)(12 20)(13 16)(14 18)(15 17)(22 25)(23 27)(24 26)(31 34)(32 36)(33 35)

G:=sub<Sym(36)| (1,25)(2,26)(3,27)(4,16)(5,17)(6,18)(7,22)(8,23)(9,24)(10,31)(11,32)(12,33)(13,28)(14,29)(15,30)(19,34)(20,35)(21,36), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,7)(2,8)(3,9)(10,13)(11,14)(12,15)(22,25)(23,26)(24,27)(28,31)(29,32)(30,33), (4,34)(5,35)(6,36)(10,13)(11,14)(12,15)(16,19)(17,20)(18,21)(28,31)(29,32)(30,33), (1,10,16)(2,11,17)(3,12,18)(4,25,31)(5,26,32)(6,27,33)(7,13,19)(8,14,20)(9,15,21)(22,28,34)(23,29,35)(24,30,36), (1,7)(2,9)(3,8)(4,28)(5,30)(6,29)(10,19)(11,21)(12,20)(13,16)(14,18)(15,17)(22,25)(23,27)(24,26)(31,34)(32,36)(33,35)>;

G:=Group( (1,25)(2,26)(3,27)(4,16)(5,17)(6,18)(7,22)(8,23)(9,24)(10,31)(11,32)(12,33)(13,28)(14,29)(15,30)(19,34)(20,35)(21,36), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36), (1,7)(2,8)(3,9)(10,13)(11,14)(12,15)(22,25)(23,26)(24,27)(28,31)(29,32)(30,33), (4,34)(5,35)(6,36)(10,13)(11,14)(12,15)(16,19)(17,20)(18,21)(28,31)(29,32)(30,33), (1,10,16)(2,11,17)(3,12,18)(4,25,31)(5,26,32)(6,27,33)(7,13,19)(8,14,20)(9,15,21)(22,28,34)(23,29,35)(24,30,36), (1,7)(2,9)(3,8)(4,28)(5,30)(6,29)(10,19)(11,21)(12,20)(13,16)(14,18)(15,17)(22,25)(23,27)(24,26)(31,34)(32,36)(33,35) );

G=PermutationGroup([(1,25),(2,26),(3,27),(4,16),(5,17),(6,18),(7,22),(8,23),(9,24),(10,31),(11,32),(12,33),(13,28),(14,29),(15,30),(19,34),(20,35),(21,36)], [(1,22),(2,23),(3,24),(4,19),(5,20),(6,21),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27),(28,29,30),(31,32,33),(34,35,36)], [(1,7),(2,8),(3,9),(10,13),(11,14),(12,15),(22,25),(23,26),(24,27),(28,31),(29,32),(30,33)], [(4,34),(5,35),(6,36),(10,13),(11,14),(12,15),(16,19),(17,20),(18,21),(28,31),(29,32),(30,33)], [(1,10,16),(2,11,17),(3,12,18),(4,25,31),(5,26,32),(6,27,33),(7,13,19),(8,14,20),(9,15,21),(22,28,34),(23,29,35),(24,30,36)], [(1,7),(2,9),(3,8),(4,28),(5,30),(6,29),(10,19),(11,21),(12,20),(13,16),(14,18),(15,17),(22,25),(23,27),(24,26),(31,34),(32,36),(33,35)])

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H ··· 6P order 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 ··· 6 size 1 1 1 1 3 3 3 3 18 18 18 18 2 8 8 8 18 18 18 18 2 2 2 6 6 6 6 8 ··· 8

36 irreducible representations

 dim 1 1 1 2 2 2 2 3 3 6 6 type + + + + + + + + + + + image C1 C2 C2 S3 S3 D6 D6 S4 C2×S4 C3⋊S4 C2×C3⋊S4 kernel C22×C3⋊S4 C2×C3⋊S4 A4×C2×C6 C22×A4 C23×C6 C2×A4 C22×C6 C2×C6 C6 C22 C2 # reps 1 6 1 3 1 9 3 2 6 1 3

Matrix representation of C22×C3⋊S4 in GL7(ℤ)

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

G:=sub<GL(7,Integers())| [-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,-1,1,0,0,0,0,0,-1,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-2,1,-1,0,0,0,0,0,0,-1] >;

C22×C3⋊S4 in GAP, Magma, Sage, TeX

C_2^2\times C_3\rtimes S_4
% in TeX

G:=Group("C2^2xC3:S4");
// GroupNames label

G:=SmallGroup(288,1034);
// by ID

G=gap.SmallGroup(288,1034);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,451,1684,6053,782,3534,1350]);
// Polycyclic

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