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## G = C22×S3×A4order 288 = 25·32

### Direct product of C22, S3 and A4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C22×S3×A4
 Chief series C1 — C3 — C2×C6 — C3×A4 — S3×A4 — C2×S3×A4 — C22×S3×A4
 Lower central C2×C6 — C22×S3×A4
 Upper central C1 — C22

Generators and relations for C22×S3×A4
G = < a,b,c,d,e,f,g | a2=b2=c3=d2=e2=f2=g3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, dcd=c-1, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >

Subgroups: 1682 in 366 conjugacy classes, 63 normal (15 characteristic)
C1, C2 [×3], C2 [×12], C3, C3 [×2], C22 [×2], C22 [×55], S3 [×4], S3 [×4], C6 [×3], C6 [×14], C23 [×3], C23 [×54], C32, A4, A4, D6 [×6], D6 [×38], C2×C6 [×2], C2×C6 [×19], C24, C24 [×14], C3×S3 [×4], C3×C6 [×3], C2×A4 [×3], C2×A4 [×7], C22×S3, C22×S3 [×4], C22×S3 [×45], C22×C6 [×3], C22×C6 [×5], C25, C3×A4, S3×C6 [×6], C62, C22×A4, C22×A4 [×7], S3×C23 [×6], S3×C23 [×8], C23×C6, S3×A4 [×4], C6×A4 [×3], S3×C2×C6, C23×A4, S3×C24, C2×S3×A4 [×6], A4×C2×C6, C22×S3×A4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, A4, D6 [×3], C2×C6 [×7], C3×S3, C2×A4 [×7], C22×S3, C22×C6, S3×C6 [×3], C22×A4 [×7], S3×A4, S3×C2×C6, C23×A4, C2×S3×A4 [×3], C22×S3×A4

Smallest permutation representation of C22×S3×A4
On 36 points
Generators in S36
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(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 22)(2 24)(3 23)(4 19)(5 21)(6 20)(7 28)(8 30)(9 29)(10 25)(11 27)(12 26)(13 34)(14 36)(15 35)(16 31)(17 33)(18 32)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2K 2L 2M 2N 2O 3A 3B 3C 3D 3E 6A 6B 6C 6D ··· 6I 6J 6K 6L 6M 6N ··· 6S 6T ··· 6AA order 1 2 2 2 2 ··· 2 2 2 2 2 3 3 3 3 3 6 6 6 6 ··· 6 6 6 6 6 6 ··· 6 6 ··· 6 size 1 1 1 1 3 ··· 3 9 9 9 9 2 4 4 8 8 2 2 2 4 ··· 4 6 6 6 6 8 ··· 8 12 ··· 12

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 3 3 3 6 6 type + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 A4 C2×A4 C2×A4 S3×A4 C2×S3×A4 kernel C22×S3×A4 C2×S3×A4 A4×C2×C6 S3×C24 S3×C23 C23×C6 C22×A4 C2×A4 C24 C23 C22×S3 D6 C2×C6 C22 C2 # reps 1 6 1 2 12 2 1 3 2 6 1 6 1 1 3

Matrix representation of C22×S3×A4 in GL7(𝔽7)

 6 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 6 0 0 0 0 0 0 0 6 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 6 0 0 0 0 0 1 6 0 0 0 0 0 0 0 6 6 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 6 0 0 0 0 0 0 6 0 0 0 0 0 0 0 6 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
,
 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 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 6
,
 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 6 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 1
,
 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 6 0 0 0 0 0 0 0 6 0

G:=sub<GL(7,GF(7))| [6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,6,6,0,0,0,0,0,0,0,6,1,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,6,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,0,0,0,0,1,0,0] >;

C22×S3×A4 in GAP, Magma, Sage, TeX

C_2^2\times S_3\times A_4
% in TeX

G:=Group("C2^2xS3xA4");
// GroupNames label

G:=SmallGroup(288,1037);
// by ID

G=gap.SmallGroup(288,1037);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-3,340,152,9414]);
// Polycyclic

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

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