metabelian, supersoluble, monomial
Aliases: C32⋊2D8, D12⋊2S3, C12.9D6, C4.8S32, (C3×C6).6D4, C3⋊2(D4⋊S3), (C3×D12)⋊1C2, C32⋊4C8⋊1C2, C6.7(C3⋊D4), (C3×C12).1C22, C2.3(D6⋊S3), SmallGroup(144,56)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊2D8
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Character table of C32⋊2D8
class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 12A | 12B | 12C | 12D | |
size | 1 | 1 | 12 | 12 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 18 | 18 | 4 | 4 | 4 | 4 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ3 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ4 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ5 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | orthogonal lifted from D4 |
ρ6 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | 0 | 0 | 1 | 1 | 0 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from D6 |
ρ7 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ8 | 2 | 2 | 0 | 2 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
ρ9 | 2 | 2 | 0 | -2 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | orthogonal lifted from D6 |
ρ10 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | √2 | -√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
ρ11 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -√2 | √2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
ρ12 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -2 | -1 | 2 | -1 | 0 | 0 | -√-3 | √-3 | 0 | 0 | -2 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
ρ13 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -2 | -1 | 2 | -1 | 0 | 0 | √-3 | -√-3 | 0 | 0 | -2 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
ρ14 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -2 | 2 | -1 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | complex lifted from C3⋊D4 |
ρ15 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -2 | 2 | -1 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | complex lifted from C3⋊D4 |
ρ16 | 4 | -4 | 0 | 0 | 4 | -2 | -2 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
ρ17 | 4 | -4 | 0 | 0 | -2 | 4 | -2 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
ρ18 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | orthogonal lifted from S32 |
ρ19 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | -4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | -1 | symplectic lifted from D6⋊S3, Schur index 2 |
ρ20 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3i | -3i | complex faithful |
ρ21 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3i | 3i | complex faithful |
(1 35 46)(2 47 36)(3 37 48)(4 41 38)(5 39 42)(6 43 40)(7 33 44)(8 45 34)(9 20 26)(10 27 21)(11 22 28)(12 29 23)(13 24 30)(14 31 17)(15 18 32)(16 25 19)
(1 46 35)(2 36 47)(3 48 37)(4 38 41)(5 42 39)(6 40 43)(7 44 33)(8 34 45)(9 20 26)(10 27 21)(11 22 28)(12 29 23)(13 24 30)(14 31 17)(15 18 32)(16 25 19)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(25 42)(26 41)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)
G:=sub<Sym(48)| (1,35,46)(2,47,36)(3,37,48)(4,41,38)(5,39,42)(6,43,40)(7,33,44)(8,45,34)(9,20,26)(10,27,21)(11,22,28)(12,29,23)(13,24,30)(14,31,17)(15,18,32)(16,25,19), (1,46,35)(2,36,47)(3,48,37)(4,38,41)(5,42,39)(6,40,43)(7,44,33)(8,34,45)(9,20,26)(10,27,21)(11,22,28)(12,29,23)(13,24,30)(14,31,17)(15,18,32)(16,25,19), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)>;
G:=Group( (1,35,46)(2,47,36)(3,37,48)(4,41,38)(5,39,42)(6,43,40)(7,33,44)(8,45,34)(9,20,26)(10,27,21)(11,22,28)(12,29,23)(13,24,30)(14,31,17)(15,18,32)(16,25,19), (1,46,35)(2,36,47)(3,48,37)(4,38,41)(5,42,39)(6,40,43)(7,44,33)(8,34,45)(9,20,26)(10,27,21)(11,22,28)(12,29,23)(13,24,30)(14,31,17)(15,18,32)(16,25,19), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43) );
G=PermutationGroup([(1,35,46),(2,47,36),(3,37,48),(4,41,38),(5,39,42),(6,43,40),(7,33,44),(8,45,34),(9,20,26),(10,27,21),(11,22,28),(12,29,23),(13,24,30),(14,31,17),(15,18,32),(16,25,19)], [(1,46,35),(2,36,47),(3,48,37),(4,38,41),(5,42,39),(6,40,43),(7,44,33),(8,34,45),(9,20,26),(10,27,21),(11,22,28),(12,29,23),(13,24,30),(14,31,17),(15,18,32),(16,25,19)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,24),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,40),(16,39),(25,42),(26,41),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43)])
C32⋊2D8 is a maximal subgroup of
C32⋊D16 C32⋊SD32 C24⋊4D6 C24⋊6D6 D12.2D6 D12.30D6 D12⋊20D6 S3×D4⋊S3 D12⋊9D6 D12⋊6D6 D12.12D6 D36⋊S3 He3⋊3D8 C33⋊6D8 C33⋊9D8
C32⋊2D8 is a maximal quotient of
C32⋊2D16 D24.S3 C32⋊2Q32 D12⋊3Dic3 C12.8Dic6 D36⋊S3 He3⋊2D8 C33⋊6D8 C33⋊9D8
Matrix representation of C32⋊2D8 ►in GL4(𝔽5) generated by
4 | 0 | 1 | 1 |
1 | 4 | 2 | 1 |
2 | 4 | 4 | 4 |
2 | 1 | 1 | 1 |
4 | 3 | 4 | 0 |
0 | 4 | 0 | 1 |
1 | 2 | 0 | 3 |
0 | 4 | 0 | 0 |
0 | 0 | 2 | 3 |
1 | 1 | 1 | 4 |
0 | 4 | 3 | 3 |
0 | 0 | 1 | 1 |
2 | 1 | 1 | 2 |
4 | 2 | 3 | 1 |
1 | 1 | 2 | 2 |
1 | 0 | 4 | 4 |
G:=sub<GL(4,GF(5))| [4,1,2,2,0,4,4,1,1,2,4,1,1,1,4,1],[4,0,1,0,3,4,2,4,4,0,0,0,0,1,3,0],[0,1,0,0,0,1,4,0,2,1,3,1,3,4,3,1],[2,4,1,1,1,2,1,0,1,3,2,4,2,1,2,4] >;
C32⋊2D8 in GAP, Magma, Sage, TeX
C_3^2\rtimes_2D_8
% in TeX
G:=Group("C3^2:2D8");
// GroupNames label
G:=SmallGroup(144,56);
// by ID
G=gap.SmallGroup(144,56);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,218,116,50,490,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C32⋊2D8 in TeX
Character table of C32⋊2D8 in TeX