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G = S3×D4⋊2S3  order 288 = 25·32

Direct product of S3 and D4⋊2S3

direct product, metabelian, supersoluble, monomial

Aliases: S3×D4⋊2S3, D12⋊11D6, Dic6⋊11D6, C62.2C23, D4⋊5S32, (S3×D4)⋊6S3, (C4×S3)⋊7D6, (C3×D4)⋊8D6, C3⋊D4⋊2D6, D12⋊5S3⋊9C2, D12⋊S3⋊9C2, (S3×C12)⋊5C22, (S3×Dic6)⋊10C2, (C2×Dic3)⋊13D6, D6.3D6⋊3C2, D6.4D6⋊5C2, (S3×C6).9C23, C6.18(S3×C23), (C3×C6).18C24, C12.D6⋊7C2, D6⋊S3⋊5C22, (C3×D12)⋊13C22, C3⋊D12⋊4C22, C12.30(C22×S3), (C3×C12).30C23, (C6×Dic3)⋊5C22, D6.10(C22×S3), (C22×S3).55D6, C32⋊2Q8⋊3C22, C6.D6⋊8C22, C32⋊7D4⋊2C22, C32⋊4Q8⋊9C22, (S3×Dic3)⋊13C22, (C3×Dic6)⋊13C22, (D4×C32)⋊10C22, C3⋊Dic3.20C23, Dic3.8(C22×S3), (C3×Dic3).12C23, (C4×S32)⋊5C2, (C3×S3×D4)⋊8C2, C4.30(C2×S32), C3⋊3(S3×C4○D4), (S3×C3⋊D4)⋊3C2, C22.2(C2×S32), (C2×S3×Dic3)⋊4C2, C32⋊7(C2×C4○D4), C3⋊3(C2×D4⋊2S3), (C4×C3⋊S3)⋊3C22, C2.20(C22×S32), (C3×S3)⋊2(C4○D4), (C3×D4⋊2S3)⋊8C2, (C2×S32).11C22, (C3×C3⋊D4)⋊2C22, (S3×C2×C6).65C22, (C2×C6).3(C22×S3), (C2×C3⋊S3).23C23, (C2×C3⋊Dic3)⋊10C22, SmallGroup(288,959)

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×D4⋊2S3
 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, D4⋊2S3, D4⋊2S3, S3×Q8, Q8⋊3S3, C22×Dic3, C2×C3⋊D4, C6×D4, C3×C4○D4, S3×Dic3, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C32⋊2Q8, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, C2×S32, S3×C2×C6, C2×D4⋊2S3, S3×C4○D4, 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⋊2S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, D4⋊2S3, S3×C23, C2×S32, C2×D4⋊2S3, S3×C4○D4, C22×S32, S3×D4⋊2S3

Smallest permutation representation of S3×D4⋊2S3
►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 2A2B2C2D2E2F2G2H2I3A3B3C4A4B4C4D4E4F4G4H4I4J6A6B6C···6G6H6I6J6K6L6M6N12A12B12C12D12E12F12G12H
order12222222223334444444444666···666666661212121212121212
size11223366618224233666991818224···4668812121244668121212

45 irreducible representations

dim1111111111112222222222444448
type++++++++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2C2C2S3S3D6D6D6D6D6D6D6C4○D4S32D4⋊2S3C2×S32C2×S32S3×C4○D4S3×D4⋊2S3
kernelS3×D4⋊2S3S3×Dic6D12⋊5S3D12⋊S3C4×S32C2×S3×Dic3D6.3D6D6.4D6S3×C3⋊D4C3×S3×D4C3×D4⋊2S3C12.D6S3×D4D4⋊2S3Dic6C4×S3D12C2×Dic3C3⋊D4C3×D4C22×S3C3×S3D4S3C4C22C3C1
# reps1111122221111112124224121221

Matrix representation of S3×D4⋊2S3 ►in GL6(𝔽13)

1210000
1200000
001000
000100
000010
000001
,
010000
100000
001000
000100
000010
000001
,
1200000
0120000
005000
000800
000010
000001
,
1200000
0120000
000500
008000
0000120
0000012
,
100000
010000
001000
000100
0000012
0000112
,
100000
010000
0012000
000100
000001
000010

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×D4⋊2S3 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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