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## G = S32×C23order 288 = 25·32

### Direct product of C23, S3 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S32×C23
 Chief series C1 — C3 — C32 — C3×S3 — S32 — C2×S32 — C22×S32 — S32×C23
 Lower central C32 — S32×C23
 Upper central C1 — C23

Generators and relations for S32×C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d3=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, cg=gc, ede=d-1, df=fd, dg=gd, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 5314 in 1563 conjugacy classes, 524 normal (6 characteristic)
C1, C2 [×7], C2 [×24], C3 [×2], C3, C22 [×7], C22 [×148], S3 [×16], S3 [×24], C6 [×14], C6 [×23], C23, C23 [×154], C32, D6 [×56], D6 [×212], C2×C6 [×14], C2×C6 [×63], C24 [×31], C3×S3 [×16], C3⋊S3 [×8], C3×C6 [×7], C22×S3 [×28], C22×S3 [×266], C22×C6 [×2], C22×C6 [×29], C25, S32 [×64], S3×C6 [×56], C2×C3⋊S3 [×28], C62 [×7], S3×C23 [×2], S3×C23 [×59], C23×C6 [×2], C2×S32 [×112], S3×C2×C6 [×28], C22×C3⋊S3 [×14], C2×C62, S3×C24 [×2], C22×S32 [×28], S3×C22×C6 [×2], C23×C3⋊S3, S32×C23
Quotients: C1, C2 [×31], C22 [×155], S3 [×2], C23 [×155], D6 [×30], C24 [×31], C22×S3 [×70], C25, S32, S3×C23 [×30], C2×S32 [×7], S3×C24 [×2], C22×S32 [×7], S32×C23

Smallest permutation representation of S32×C23
On 48 points
Generators in S48
(1 20)(2 21)(3 19)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 5)(2 6)(3 4)(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)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)
(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)(37 38 39)(40 41 42)(43 44 45)(46 47 48)
(1 37)(2 39)(3 38)(4 41)(5 40)(6 42)(7 44)(8 43)(9 45)(10 47)(11 46)(12 48)(13 26)(14 25)(15 27)(16 29)(17 28)(18 30)(19 32)(20 31)(21 33)(22 35)(23 34)(24 36)
(1 3 2)(4 6 5)(7 9 8)(10 12 11)(13 15 14)(16 18 17)(19 21 20)(22 24 23)(25 26 27)(28 29 30)(31 32 33)(34 35 36)(37 38 39)(40 41 42)(43 44 45)(46 47 48)
(1 43)(2 44)(3 45)(4 48)(5 46)(6 47)(7 39)(8 37)(9 38)(10 42)(11 40)(12 41)(13 33)(14 31)(15 32)(16 36)(17 34)(18 35)(19 27)(20 25)(21 26)(22 30)(23 28)(24 29)

G:=sub<Sym(48)| (1,20)(2,21)(3,19)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (1,8)(2,9)(3,7)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,5)(2,6)(3,4)(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)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48), (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)(37,38,39)(40,41,42)(43,44,45)(46,47,48), (1,37)(2,39)(3,38)(4,41)(5,40)(6,42)(7,44)(8,43)(9,45)(10,47)(11,46)(12,48)(13,26)(14,25)(15,27)(16,29)(17,28)(18,30)(19,32)(20,31)(21,33)(22,35)(23,34)(24,36), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,41,42)(43,44,45)(46,47,48), (1,43)(2,44)(3,45)(4,48)(5,46)(6,47)(7,39)(8,37)(9,38)(10,42)(11,40)(12,41)(13,33)(14,31)(15,32)(16,36)(17,34)(18,35)(19,27)(20,25)(21,26)(22,30)(23,28)(24,29)>;

G:=Group( (1,20)(2,21)(3,19)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (1,8)(2,9)(3,7)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,5)(2,6)(3,4)(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)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48), (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)(37,38,39)(40,41,42)(43,44,45)(46,47,48), (1,37)(2,39)(3,38)(4,41)(5,40)(6,42)(7,44)(8,43)(9,45)(10,47)(11,46)(12,48)(13,26)(14,25)(15,27)(16,29)(17,28)(18,30)(19,32)(20,31)(21,33)(22,35)(23,34)(24,36), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,41,42)(43,44,45)(46,47,48), (1,43)(2,44)(3,45)(4,48)(5,46)(6,47)(7,39)(8,37)(9,38)(10,42)(11,40)(12,41)(13,33)(14,31)(15,32)(16,36)(17,34)(18,35)(19,27)(20,25)(21,26)(22,30)(23,28)(24,29) );

G=PermutationGroup([(1,20),(2,21),(3,19),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42)], [(1,8),(2,9),(3,7),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,5),(2,6),(3,4),(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),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48)], [(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),(37,38,39),(40,41,42),(43,44,45),(46,47,48)], [(1,37),(2,39),(3,38),(4,41),(5,40),(6,42),(7,44),(8,43),(9,45),(10,47),(11,46),(12,48),(13,26),(14,25),(15,27),(16,29),(17,28),(18,30),(19,32),(20,31),(21,33),(22,35),(23,34),(24,36)], [(1,3,2),(4,6,5),(7,9,8),(10,12,11),(13,15,14),(16,18,17),(19,21,20),(22,24,23),(25,26,27),(28,29,30),(31,32,33),(34,35,36),(37,38,39),(40,41,42),(43,44,45),(46,47,48)], [(1,43),(2,44),(3,45),(4,48),(5,46),(6,47),(7,39),(8,37),(9,38),(10,42),(11,40),(12,41),(13,33),(14,31),(15,32),(16,36),(17,34),(18,35),(19,27),(20,25),(21,26),(22,30),(23,28),(24,29)])

72 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2W 2X ··· 2AE 3A 3B 3C 6A ··· 6N 6O ··· 6U 6V ··· 6AK order 1 2 ··· 2 2 ··· 2 2 ··· 2 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 ··· 1 3 ··· 3 9 ··· 9 2 2 4 2 ··· 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 S3 D6 D6 S32 C2×S32 kernel S32×C23 C22×S32 S3×C22×C6 C23×C3⋊S3 S3×C23 C22×S3 C22×C6 C23 C22 # reps 1 28 2 1 2 28 2 1 7

Matrix representation of S32×C23 in GL8(ℤ)

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

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

S32×C23 in GAP, Magma, Sage, TeX

S_3^2\times C_2^3
% in TeX

G:=Group("S3^2xC2^3");
// GroupNames label

G:=SmallGroup(288,1040);
// by ID

G=gap.SmallGroup(288,1040);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,1356,9414]);
// Polycyclic

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

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