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## G = S3×D4⋊2S3order 288 = 25·32

### Direct product of S3 and D4⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S3×D4⋊2S3
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — C4×S32 — S3×D4⋊2S3
 Lower central C32 — C3×C6 — S3×D4⋊2S3
 Upper central C1 — C2 — D4

Generators and relations for S3×D42S3
G = < a,b,c,d,e,f | a3=b2=c4=d2=e3=f2=1, 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, fdf=c2d, fef=e-1 >

Subgroups: 1178 in 348 conjugacy classes, 112 normal (50 characteristic)
C1, C2, C2 [×8], C3 [×2], C3, C4, C4 [×7], C22 [×2], C22 [×11], S3 [×2], S3 [×6], C6 [×2], C6 [×12], C2×C4 [×16], D4, D4 [×11], Q8 [×4], C23 [×3], C32, Dic3 [×2], Dic3 [×2], Dic3 [×9], C12 [×2], C12 [×5], D6 [×2], D6 [×2], D6 [×11], C2×C6 [×4], C2×C6 [×10], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×2], C3×S3 [×3], C3⋊S3, C3×C6, C3×C6 [×2], Dic6, Dic6 [×7], C4×S3 [×2], C4×S3 [×13], D12, D12 [×2], C2×Dic3 [×2], C2×Dic3 [×14], C3⋊D4 [×4], C3⋊D4 [×12], C2×C12 [×4], C3×D4 [×2], C3×D4 [×6], C3×Q8, C22×S3 [×2], C22×S3 [×2], C22×C6 [×2], C2×C4○D4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3, C3⋊Dic3 [×2], C3×C12, S32 [×2], S3×C6 [×2], S3×C6 [×2], S3×C6 [×4], C2×C3⋊S3, C62 [×2], C2×Dic6, S3×C2×C4 [×4], C4○D12 [×3], S3×D4, S3×D4 [×2], D42S3, D42S3 [×11], S3×Q8, Q83S3, C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C3×C4○D4, S3×Dic3 [×2], S3×Dic3 [×6], C6.D6, D6⋊S3 [×2], C3⋊D12 [×2], C322Q8 [×2], C3×Dic6, S3×C12 [×2], C3×D12, C6×Dic3 [×2], C3×C3⋊D4 [×4], C324Q8, C4×C3⋊S3, C2×C3⋊Dic3 [×2], C327D4 [×2], D4×C32, C2×S32, S3×C2×C6 [×2], C2×D42S3, S3×C4○D4, S3×Dic6, D125S3, D12⋊S3, C4×S32, C2×S3×Dic3 [×2], D6.3D6 [×2], D6.4D6 [×2], S3×C3⋊D4 [×2], C3×S3×D4, C3×D42S3, C12.D6, S3×D42S3
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, D42S3 [×2], S3×C23 [×2], C2×S32 [×3], C2×D42S3, S3×C4○D4, C22×S32, S3×D42S3

Smallest permutation representation of S3×D42S3
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 43 47)(14 44 48)(15 41 45)(16 42 46)(17 21 27)(18 22 28)(19 23 25)(20 24 26)(29 35 38)(30 36 39)(31 33 40)(32 34 37)
(1 33)(2 34)(3 35)(4 36)(5 40)(6 37)(7 38)(8 39)(9 31)(10 32)(11 29)(12 30)(13 23)(14 24)(15 21)(16 22)(17 41)(18 42)(19 43)(20 44)(25 47)(26 48)(27 45)(28 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 20)(2 19)(3 18)(4 17)(5 26)(6 25)(7 28)(8 27)(9 24)(10 23)(11 22)(12 21)(13 32)(14 31)(15 30)(16 29)(33 44)(34 43)(35 42)(36 41)(37 47)(38 46)(39 45)(40 48)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 47 43)(14 48 44)(15 45 41)(16 46 42)(17 21 27)(18 22 28)(19 23 25)(20 24 26)(29 38 35)(30 39 36)(31 40 33)(32 37 34)
(1 35)(2 36)(3 33)(4 34)(5 29)(6 30)(7 31)(8 32)(9 38)(10 39)(11 40)(12 37)(13 25)(14 26)(15 27)(16 28)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)

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

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

G=PermutationGroup([(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,43,47),(14,44,48),(15,41,45),(16,42,46),(17,21,27),(18,22,28),(19,23,25),(20,24,26),(29,35,38),(30,36,39),(31,33,40),(32,34,37)], [(1,33),(2,34),(3,35),(4,36),(5,40),(6,37),(7,38),(8,39),(9,31),(10,32),(11,29),(12,30),(13,23),(14,24),(15,21),(16,22),(17,41),(18,42),(19,43),(20,44),(25,47),(26,48),(27,45),(28,46)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,26),(6,25),(7,28),(8,27),(9,24),(10,23),(11,22),(12,21),(13,32),(14,31),(15,30),(16,29),(33,44),(34,43),(35,42),(36,41),(37,47),(38,46),(39,45),(40,48)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,47,43),(14,48,44),(15,45,41),(16,46,42),(17,21,27),(18,22,28),(19,23,25),(20,24,26),(29,38,35),(30,39,36),(31,40,33),(32,37,34)], [(1,35),(2,36),(3,33),(4,34),(5,29),(6,30),(7,31),(8,32),(9,38),(10,39),(11,40),(12,37),(13,25),(14,26),(15,27),(16,28),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(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 ··· 6G 6H 6I 6J 6K 6L 6M 6N 12A 12B 12C 12D 12E 12F 12G 12H 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 6 6 6 12 12 12 12 12 12 12 12 size 1 1 2 2 3 3 6 6 6 18 2 2 4 2 3 3 6 6 6 9 9 18 18 2 2 4 ··· 4 6 6 8 8 12 12 12 4 4 6 6 8 12 12 12

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 S3 D6 D6 D6 D6 D6 D6 D6 C4○D4 S32 D4⋊2S3 C2×S32 C2×S32 S3×C4○D4 S3×D4⋊2S3 kernel S3×D4⋊2S3 S3×Dic6 D12⋊5S3 D12⋊S3 C4×S32 C2×S3×Dic3 D6.3D6 D6.4D6 S3×C3⋊D4 C3×S3×D4 C3×D4⋊2S3 C12.D6 S3×D4 D4⋊2S3 Dic6 C4×S3 D12 C2×Dic3 C3⋊D4 C3×D4 C22×S3 C3×S3 D4 S3 C4 C22 C3 C1 # reps 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 2 1 2 4 2 2 4 1 2 1 2 2 1

Matrix representation of S3×D42S3 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 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 5 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

S3×D42S3 in GAP, Magma, Sage, TeX

S_3\times D_4\rtimes_2S_3
% in TeX

G:=Group("S3xD4:2S3");
// GroupNames label

G:=SmallGroup(288,959);
// by ID

G=gap.SmallGroup(288,959);
# by ID

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

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

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