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
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 [×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], D4⋊2S3, D4⋊2S3 [×11], S3×Q8, Q8⋊3S3, 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], C32⋊2Q8 [×2], C3×Dic6, S3×C12 [×2], C3×D12, C6×Dic3 [×2], C3×C3⋊D4 [×4], C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3 [×2], C32⋊7D4 [×2], D4×C32, C2×S32, S3×C2×C6 [×2], C2×D4⋊2S3, S3×C4○D4, S3×Dic6, D12⋊5S3, D12⋊S3, C4×S32, C2×S3×Dic3 [×2], D6.3D6 [×2], D6.4D6 [×2], S3×C3⋊D4 [×2], C3×S3×D4, C3×D4⋊2S3, C12.D6, S3×D4⋊2S3
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, D4⋊2S3 [×2], S3×C23 [×2], C2×S32 [×3], C2×D4⋊2S3, S3×C4○D4, C22×S32, S3×D4⋊2S3
(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×D4⋊2S3 ►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×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