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## G = S32×C2×C4order 288 = 25·32

### Direct product of C2×C4, S3 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S32×C2×C4
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C22×S32 — S32×C2×C4
 Lower central C32 — S32×C2×C4
 Upper central C1 — C2×C4

Generators and relations for S32×C2×C4
G = < a,b,c,d,e,f | a2=b4=c3=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1618 in 499 conjugacy classes, 180 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×12], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×34], S3 [×8], S3 [×12], C6 [×6], C6 [×11], C2×C4, C2×C4 [×27], C23 [×15], C32, Dic3 [×4], Dic3 [×6], C12 [×4], C12 [×6], D6 [×12], D6 [×50], C2×C6 [×2], C2×C6 [×13], C22×C4 [×14], C24, C3×S3 [×8], C3⋊S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×8], C4×S3 [×28], C2×Dic3 [×2], C2×Dic3 [×11], C2×C12 [×2], C2×C12 [×11], C22×S3 [×2], C22×S3 [×27], C22×C6 [×2], C23×C4, C3×Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], S32 [×16], S3×C6 [×12], C2×C3⋊S3 [×6], C62, S3×C2×C4 [×2], S3×C2×C4 [×23], C22×Dic3 [×2], C22×C12 [×2], S3×C23 [×2], S3×Dic3 [×8], C6.D6 [×4], S3×C12 [×8], C6×Dic3 [×2], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C2×S32 [×12], S3×C2×C6 [×2], C22×C3⋊S3, S3×C22×C4 [×2], C4×S32 [×8], C2×S3×Dic3 [×2], C2×C6.D6, S3×C2×C12 [×2], C2×C4×C3⋊S3, C22×S32, S32×C2×C4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3 [×2], C2×C4 [×28], C23 [×15], D6 [×14], C22×C4 [×14], C24, C4×S3 [×8], C22×S3 [×14], C23×C4, S32, S3×C2×C4 [×12], S3×C23 [×2], C2×S32 [×3], S3×C22×C4 [×2], C4×S32 [×2], C22×S32, S32×C2×C4

Smallest permutation representation of S32×C2×C4
On 48 points
Generators in S48
(1 22)(2 23)(3 24)(4 21)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(33 45)(34 46)(35 47)(36 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 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 39 44)(34 40 41)(35 37 42)(36 38 43)
(1 46)(2 47)(3 48)(4 45)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 27 32)(22 28 29)(23 25 30)(24 26 31)(33 39 44)(34 40 41)(35 37 42)(36 38 43)
(1 36)(2 33)(3 34)(4 35)(5 32)(6 29)(7 30)(8 31)(9 26)(10 27)(11 28)(12 25)(13 44)(14 41)(15 42)(16 43)(17 38)(18 39)(19 40)(20 37)(21 47)(22 48)(23 45)(24 46)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2K 2L 2M 2N 2O 3A 3B 3C 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 4P 6A ··· 6F 6G 6H 6I 6J ··· 6Q 12A ··· 12H 12I 12J 12K 12L 12M ··· 12T order 1 2 2 2 2 ··· 2 2 2 2 2 3 3 3 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 6 ··· 6 12 ··· 12 12 12 12 12 12 ··· 12 size 1 1 1 1 3 ··· 3 9 9 9 9 2 2 4 1 1 1 1 3 ··· 3 9 9 9 9 2 ··· 2 4 4 4 6 ··· 6 2 ··· 2 4 4 4 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 S3 D6 D6 D6 D6 C4×S3 S32 C2×S32 C2×S32 C4×S32 kernel S32×C2×C4 C4×S32 C2×S3×Dic3 C2×C6.D6 S3×C2×C12 C2×C4×C3⋊S3 C22×S32 C2×S32 S3×C2×C4 C4×S3 C2×Dic3 C2×C12 C22×S3 D6 C2×C4 C4 C22 C2 # reps 1 8 2 1 2 1 1 16 2 8 2 2 2 16 1 2 1 4

Matrix representation of S32×C2×C4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 0 8 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 12 1 0 0 0 0 12 0
,
 12 0 0 0 0 0 0 12 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
,
 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S32×C2×C4 in GAP, Magma, Sage, TeX

S_3^2\times C_2\times C_4
% in TeX

G:=Group("S3^2xC2xC4");
// GroupNames label

G:=SmallGroup(288,950);
// by ID

G=gap.SmallGroup(288,950);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^4=c^3=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,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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