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## G = S3×Q8⋊3S3order 288 = 25·32

### Direct product of S3 and Q8⋊3S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S3×Q8⋊3S3
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C4×S32 — S3×Q8⋊3S3
 Lower central C32 — C3×C6 — S3×Q8⋊3S3
 Upper central C1 — C2 — Q8

Generators and relations for S3×Q83S3
G = < a,b,c,d,e,f | a3=b2=c4=e3=f2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fcf=c-1, ce=ec, de=ed, df=fd, fef=e-1 >

Subgroups: 1306 in 348 conjugacy classes, 112 normal (24 characteristic)
C1, C2, C2 [×8], C3 [×2], C3, C4 [×3], C4 [×5], C22 [×13], S3 [×2], S3 [×12], C6 [×2], C6 [×6], C2×C4 [×16], D4 [×12], Q8, Q8 [×3], C23 [×3], C32, Dic3, Dic3 [×3], Dic3 [×3], C12 [×6], C12 [×7], D6, D6 [×3], D6 [×21], C2×C6 [×4], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×2], C3×S3 [×3], C3⋊S3 [×3], C3×C6, Dic6 [×3], C4×S3 [×6], C4×S3 [×19], D12 [×3], D12 [×15], C2×Dic3 [×4], C3⋊D4 [×6], C2×C12 [×6], C3×D4 [×3], C3×Q8 [×2], C3×Q8 [×4], C22×S3 [×6], C2×C4○D4, C3×Dic3, C3×Dic3 [×3], C3⋊Dic3, C3×C12 [×3], S32 [×6], S3×C6, S3×C6 [×3], C2×C3⋊S3 [×3], S3×C2×C4 [×6], C2×D12 [×3], C4○D12 [×3], S3×D4 [×3], D42S3 [×3], S3×Q8, Q83S3, Q83S3 [×9], C6×Q8, C3×C4○D4, S3×Dic3, S3×Dic3 [×3], C6.D6 [×3], C3⋊D12 [×6], C3×Dic6 [×3], S3×C12 [×6], C3×D12 [×3], C4×C3⋊S3 [×3], C12⋊S3 [×3], Q8×C32, C2×S32 [×3], C2×Q83S3, S3×C4○D4, D12⋊S3 [×3], D6.6D6 [×3], C4×S32 [×3], S3×D12 [×3], C3×S3×Q8, C3×Q83S3, C12.26D6, S3×Q83S3
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, Q83S3 [×2], S3×C23 [×2], C2×S32 [×3], C2×Q83S3, S3×C4○D4, C22×S32, S3×Q83S3

Smallest permutation representation of S3×Q83S3
On 48 points
Generators in S48
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(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 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 33)(2 34)(3 35)(4 36)(5 28)(6 25)(7 26)(8 27)(9 32)(10 29)(11 30)(12 31)(13 40)(14 37)(15 38)(16 39)(17 44)(18 41)(19 42)(20 43)(21 46)(22 47)(23 48)(24 45)
(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 37 7 39)(6 40 8 38)(9 43 11 41)(10 42 12 44)(13 27 15 25)(14 26 16 28)(17 29 19 31)(18 32 20 30)(33 45 35 47)(34 48 36 46)
(1 19 14)(2 20 15)(3 17 16)(4 18 13)(5 47 10)(6 48 11)(7 45 12)(8 46 9)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 42 37)(34 43 38)(35 44 39)(36 41 40)
(1 34)(2 33)(3 36)(4 35)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 47)(22 46)(23 45)(24 48)

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 C4○D4 S32 Q8⋊3S3 C2×S32 S3×C4○D4 S3×Q8⋊3S3 kernel S3×Q8⋊3S3 D12⋊S3 D6.6D6 C4×S32 S3×D12 C3×S3×Q8 C3×Q8⋊3S3 C12.26D6 S3×Q8 Q8⋊3S3 Dic6 C4×S3 D12 C3×Q8 C3×S3 Q8 S3 C4 C3 C1 # reps 1 3 3 3 3 1 1 1 1 1 3 6 3 2 4 1 2 3 2 1

Matrix representation of S3×Q83S3 in GL6(𝔽13)

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

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

S3×Q83S3 in GAP, Magma, Sage, TeX

S_3\times Q_8\rtimes_3S_3
% in TeX

G:=Group("S3xQ8:3S3");
// GroupNames label

G:=SmallGroup(288,966);
// by ID

G=gap.SmallGroup(288,966);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,346,185,80,1356,9414]);
// Polycyclic

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

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