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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, C3, C3, C4, C4, C22, C22, S3, S3, C6, C6, C2×C4, D4, D4, Q8, C23, C32, Dic3, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C2×C4○D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, S3×C6, C2×C3⋊S3, C62, C2×Dic6, S3×C2×C4, C4○D12, S3×D4, S3×D4, D42S3, D42S3, S3×Q8, Q83S3, C22×Dic3, C2×C3⋊D4, C6×D4, C3×C4○D4, S3×Dic3, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C322Q8, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C327D4, D4×C32, C2×S32, S3×C2×C6, C2×D42S3, S3×C4○D4, S3×Dic6, D125S3, D12⋊S3, C4×S32, C2×S3×Dic3, D6.3D6, D6.4D6, S3×C3⋊D4, C3×S3×D4, C3×D42S3, C12.D6, S3×D42S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, D42S3, S3×C23, C2×S32, C2×D42S3, S3×C4○D4, C22×S32, S3×D42S3

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

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

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

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