metabelian, supersoluble, monomial
Aliases: D6⋊3S32, Dic3⋊3S32, C33⋊8(C2×D4), (S3×C6)⋊10D6, C3⋊D12⋊3S3, (C3×Dic3)⋊1D6, C32⋊16(S3×D4), C33⋊6D4⋊2C2, C3⋊4(D6⋊D6), C33⋊5C4⋊4C22, (C32×C6).7C23, (C32×Dic3)⋊2C22, C2.7S33, (S32×C6)⋊4C2, (C2×S32)⋊4S3, C6.7(C2×S32), (C2×C3⋊S3)⋊9D6, (C3×C3⋊S3)⋊3D4, C3⋊3(S3×C3⋊D4), (S3×C3×C6)⋊4C22, C3⋊S3⋊3(C3⋊D4), (C6×C3⋊S3)⋊3C22, (Dic3×C3⋊S3)⋊1C2, C32⋊8(C2×C3⋊D4), (C3×C3⋊D12)⋊2C2, (C2×C32⋊4D6)⋊1C2, (C3×C6).56(C22×S3), SmallGroup(432,600)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊S32
G = < a,b,c,d,e,f | a3=b3=c3=d2=e4=f2=1, ab=ba, ac=ca, ad=da, eae-1=a-1, af=fa, bc=cb, dbd=ebe-1=fbf=b-1, dcd=ece-1=c-1, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 1676 in 270 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C22×S3, C22×C6, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×D4, C2×C3⋊D4, S3×C32, C3×C3⋊S3, C3×C3⋊S3, C32×C6, S3×Dic3, D6⋊S3, C3⋊D12, C3×D12, C3×C3⋊D4, C4×C3⋊S3, C32⋊7D4, C2×S32, C2×S32, S3×C2×C6, C32×Dic3, C33⋊5C4, C3×S32, C32⋊4D6, S3×C3×C6, C6×C3⋊S3, C6×C3⋊S3, D6⋊D6, S3×C3⋊D4, C3×C3⋊D12, Dic3×C3⋊S3, C33⋊6D4, S32×C6, C2×C32⋊4D6, D6⋊S32
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, S32, S3×D4, C2×C3⋊D4, C2×S32, D6⋊D6, S3×C3⋊D4, S33, D6⋊S32
►On 48 points
(1 44 45)(2 46 41)(3 42 47)(4 48 43)(5 28 24)(6 21 25)(7 26 22)(8 23 27)(9 37 30)(10 31 38)(11 39 32)(12 29 40)(13 18 35)(14 36 19)(15 20 33)(16 34 17)
(1 44 45)(2 46 41)(3 42 47)(4 48 43)(5 24 28)(6 25 21)(7 22 26)(8 27 23)(9 30 37)(10 38 31)(11 32 39)(12 40 29)(13 18 35)(14 36 19)(15 20 33)(16 34 17)
(1 45 44)(2 41 46)(3 47 42)(4 43 48)(5 28 24)(6 21 25)(7 26 22)(8 23 27)(9 30 37)(10 38 31)(11 32 39)(12 40 29)(13 18 35)(14 36 19)(15 20 33)(16 34 17)
(1 6)(2 7)(3 8)(4 5)(9 19)(10 20)(11 17)(12 18)(13 40)(14 37)(15 38)(16 39)(21 44)(22 41)(23 42)(24 43)(25 45)(26 46)(27 47)(28 48)(29 35)(30 36)(31 33)(32 34)
(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 30)(2 29)(3 32)(4 31)(5 33)(6 36)(7 35)(8 34)(9 44)(10 43)(11 42)(12 41)(13 26)(14 25)(15 28)(16 27)(17 23)(18 22)(19 21)(20 24)(37 45)(38 48)(39 47)(40 46)
G:=sub<Sym(48)| (1,44,45)(2,46,41)(3,42,47)(4,48,43)(5,28,24)(6,21,25)(7,26,22)(8,23,27)(9,37,30)(10,31,38)(11,39,32)(12,29,40)(13,18,35)(14,36,19)(15,20,33)(16,34,17), (1,44,45)(2,46,41)(3,42,47)(4,48,43)(5,24,28)(6,25,21)(7,22,26)(8,27,23)(9,30,37)(10,38,31)(11,32,39)(12,40,29)(13,18,35)(14,36,19)(15,20,33)(16,34,17), (1,45,44)(2,41,46)(3,47,42)(4,43,48)(5,28,24)(6,21,25)(7,26,22)(8,23,27)(9,30,37)(10,38,31)(11,32,39)(12,40,29)(13,18,35)(14,36,19)(15,20,33)(16,34,17), (1,6)(2,7)(3,8)(4,5)(9,19)(10,20)(11,17)(12,18)(13,40)(14,37)(15,38)(16,39)(21,44)(22,41)(23,42)(24,43)(25,45)(26,46)(27,47)(28,48)(29,35)(30,36)(31,33)(32,34), (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), (1,30)(2,29)(3,32)(4,31)(5,33)(6,36)(7,35)(8,34)(9,44)(10,43)(11,42)(12,41)(13,26)(14,25)(15,28)(16,27)(17,23)(18,22)(19,21)(20,24)(37,45)(38,48)(39,47)(40,46)>;
G:=Group( (1,44,45)(2,46,41)(3,42,47)(4,48,43)(5,28,24)(6,21,25)(7,26,22)(8,23,27)(9,37,30)(10,31,38)(11,39,32)(12,29,40)(13,18,35)(14,36,19)(15,20,33)(16,34,17), (1,44,45)(2,46,41)(3,42,47)(4,48,43)(5,24,28)(6,25,21)(7,22,26)(8,27,23)(9,30,37)(10,38,31)(11,32,39)(12,40,29)(13,18,35)(14,36,19)(15,20,33)(16,34,17), (1,45,44)(2,41,46)(3,47,42)(4,43,48)(5,28,24)(6,21,25)(7,26,22)(8,23,27)(9,30,37)(10,38,31)(11,32,39)(12,40,29)(13,18,35)(14,36,19)(15,20,33)(16,34,17), (1,6)(2,7)(3,8)(4,5)(9,19)(10,20)(11,17)(12,18)(13,40)(14,37)(15,38)(16,39)(21,44)(22,41)(23,42)(24,43)(25,45)(26,46)(27,47)(28,48)(29,35)(30,36)(31,33)(32,34), (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), (1,30)(2,29)(3,32)(4,31)(5,33)(6,36)(7,35)(8,34)(9,44)(10,43)(11,42)(12,41)(13,26)(14,25)(15,28)(16,27)(17,23)(18,22)(19,21)(20,24)(37,45)(38,48)(39,47)(40,46) );
G=PermutationGroup([[(1,44,45),(2,46,41),(3,42,47),(4,48,43),(5,28,24),(6,21,25),(7,26,22),(8,23,27),(9,37,30),(10,31,38),(11,39,32),(12,29,40),(13,18,35),(14,36,19),(15,20,33),(16,34,17)], [(1,44,45),(2,46,41),(3,42,47),(4,48,43),(5,24,28),(6,25,21),(7,22,26),(8,27,23),(9,30,37),(10,38,31),(11,32,39),(12,40,29),(13,18,35),(14,36,19),(15,20,33),(16,34,17)], [(1,45,44),(2,41,46),(3,47,42),(4,43,48),(5,28,24),(6,21,25),(7,26,22),(8,23,27),(9,30,37),(10,38,31),(11,32,39),(12,40,29),(13,18,35),(14,36,19),(15,20,33),(16,34,17)], [(1,6),(2,7),(3,8),(4,5),(9,19),(10,20),(11,17),(12,18),(13,40),(14,37),(15,38),(16,39),(21,44),(22,41),(23,42),(24,43),(25,45),(26,46),(27,47),(28,48),(29,35),(30,36),(31,33),(32,34)], [(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)], [(1,30),(2,29),(3,32),(4,31),(5,33),(6,36),(7,35),(8,34),(9,44),(10,43),(11,42),(12,41),(13,26),(14,25),(15,28),(16,27),(17,23),(18,22),(19,21),(20,24),(37,45),(38,48),(39,47),(40,46)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | ··· | 6Q | 6R | 6S | 6T | 6U | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 6 | 6 | 9 | 9 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 6 | 54 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 12 | ··· | 12 | 18 | 18 | 36 | 36 | 12 | 12 | 12 | 12 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D6 | C3⋊D4 | S32 | S32 | S3×D4 | C2×S32 | D6⋊D6 | S3×C3⋊D4 | S33 | D6⋊S32 |
| kernel | D6⋊S32 | C3×C3⋊D12 | Dic3×C3⋊S3 | C33⋊6D4 | S32×C6 | C2×C32⋊4D6 | C3⋊D12 | C2×S32 | C3×C3⋊S3 | C3×Dic3 | S3×C6 | C2×C3⋊S3 | C3⋊S3 | Dic3 | D6 | C32 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 3 | 4 | 1 | 2 | 2 | 3 | 2 | 4 | 1 | 1 |
Matrix representation of D6⋊S32 ►in GL8(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;
D6⋊S32 in GAP, Magma, Sage, TeX
D_6\rtimes S_3^2
% in TeX
G:=Group("D6:S3^2"); // GroupNames label
G:=SmallGroup(432,600);
// by ID
G=gap.SmallGroup(432,600);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,58,298,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,a*f=f*a,b*c=c*b,d*b*d=e*b*e^-1=f*b*f=b^-1,d*c*d=e*c*e^-1=c^-1,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations