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## G = S32×Q8order 288 = 25·32

### Direct product of S3, S3 and Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S32×Q8
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C4×S32 — S32×Q8
 Lower central C32 — C3×C6 — S32×Q8
 Upper central C1 — C2 — Q8

Generators and relations for S32×Q8
G = < a,b,c,d,e,f | a3=b2=c3=d2=e4=1, f2=e2, bab=a-1, 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-1=e-1 >

Subgroups: 1058 in 331 conjugacy classes, 122 normal (10 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4 [×3], C4 [×9], C22 [×7], S3 [×4], S3 [×6], C6 [×2], C6 [×5], C2×C4 [×18], Q8, Q8 [×15], C23, C32, Dic3 [×6], Dic3 [×9], C12 [×6], C12 [×9], D6 [×2], D6 [×11], C2×C6 [×2], C22×C4 [×3], C2×Q8 [×12], C3×S3 [×4], C3⋊S3 [×2], C3×C6, Dic6 [×6], Dic6 [×21], C4×S3 [×6], C4×S3 [×21], C2×Dic3 [×6], C2×C12 [×6], C3×Q8 [×2], C3×Q8 [×7], C22×S3 [×2], C22×Q8, C3×Dic3 [×6], C3⋊Dic3 [×3], C3×C12 [×3], S32 [×4], S3×C6 [×2], C2×C3⋊S3, C2×Dic6 [×6], S3×C2×C4 [×6], S3×Q8 [×2], S3×Q8 [×15], C6×Q8 [×2], S3×Dic3 [×6], C6.D6 [×3], C322Q8 [×6], C3×Dic6 [×6], S3×C12 [×6], C324Q8 [×3], C4×C3⋊S3 [×3], Q8×C32, C2×S32, C2×S3×Q8 [×2], S3×Dic6 [×6], Dic3.D6 [×3], C4×S32 [×3], C3×S3×Q8 [×2], Q8×C3⋊S3, S32×Q8
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], Q8 [×4], C23 [×15], D6 [×14], C2×Q8 [×6], C24, C22×S3 [×14], C22×Q8, S32, S3×Q8 [×4], S3×C23 [×2], C2×S32 [×3], C2×S3×Q8 [×2], C22×S32, S32×Q8

Smallest permutation representation of S32×Q8
On 48 points
Generators in S48
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 10 45)(6 11 46)(7 12 47)(8 9 48)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 42 37)(34 43 38)(35 44 39)(36 41 40)
(1 34)(2 35)(3 36)(4 33)(5 30)(6 31)(7 32)(8 29)(9 28)(10 25)(11 26)(12 27)(13 42)(14 43)(15 44)(16 41)(17 40)(18 37)(19 38)(20 39)(21 45)(22 46)(23 47)(24 48)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 45)(6 11 46)(7 12 47)(8 9 48)(21 30 25)(22 31 26)(23 32 27)(24 29 28)(33 42 37)(34 43 38)(35 44 39)(36 41 40)
(1 36)(2 33)(3 34)(4 35)(5 27)(6 28)(7 25)(8 26)(9 31)(10 32)(11 29)(12 30)(13 39)(14 40)(15 37)(16 38)(17 43)(18 44)(19 41)(20 42)(21 47)(22 48)(23 45)(24 46)
(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 24 3 22)(2 23 4 21)(5 44 7 42)(6 43 8 41)(9 40 11 38)(10 39 12 37)(13 30 15 32)(14 29 16 31)(17 26 19 28)(18 25 20 27)(33 45 35 47)(34 48 36 46)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D ··· 4I 4J 4K 4L 6A 6B 6C 6D 6E 6F 6G 12A ··· 12F 12G 12H 12I 12J ··· 12O order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 ··· 4 4 4 4 6 6 6 6 6 6 6 12 ··· 12 12 12 12 12 ··· 12 size 1 1 3 3 3 3 9 9 2 2 4 2 2 2 6 ··· 6 18 18 18 2 2 4 6 6 6 6 4 ··· 4 8 8 8 12 ··· 12

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 8 type + + + + + + + - + + + + - + - image C1 C2 C2 C2 C2 C2 S3 Q8 D6 D6 D6 S32 S3×Q8 C2×S32 S32×Q8 kernel S32×Q8 S3×Dic6 Dic3.D6 C4×S32 C3×S3×Q8 Q8×C3⋊S3 S3×Q8 S32 Dic6 C4×S3 C3×Q8 Q8 S3 C4 C1 # reps 1 6 3 3 2 1 2 4 6 6 2 1 4 3 1

Matrix representation of S32×Q8 in GL6(𝔽13)

 12 1 0 0 0 0 12 0 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 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 0 1 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 1 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 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 11 0 0 0 0 1 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 10 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,11,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,10,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S32×Q8 in GAP, Magma, Sage, TeX

S_3^2\times Q_8
% in TeX

G:=Group("S3^2xQ8");
// GroupNames label

G:=SmallGroup(288,965);
// by ID

G=gap.SmallGroup(288,965);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,185,80,1356,9414]);
// Polycyclic

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

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