p-group, metabelian, nilpotent (class 3), monomial
Aliases: D8⋊6D4, Q16⋊6D4, SD16○SD16, SD16⋊12D4, C42.456C23, M4(2).16C23, 2+ 1+4⋊3C22, C22.42+ 1+4, 2- 1+4.4C22, C2.75D42, C8○D8⋊9C2, C8.12(C2×D4), D4⋊4D4⋊6C2, C8⋊5D4⋊10C2, (C4×C8)⋊34C22, D4○SD16⋊4C2, D4.34(C2×D4), C8○D4⋊5C22, C4⋊Q8⋊19C22, C4≀C2⋊13C22, Q8.34(C2×D4), D4.3D4⋊6C2, (C2×C4).24C24, D4.10D4⋊6C2, (C2×Q8).8C23, (C2×C8).288C23, C4○D4.13C23, C4○D8.28C22, (C2×D4).10C23, C4.105(C22×D4), C4.D4⋊4C22, C8⋊C22.2C22, C8.C22⋊4C22, C8.C4⋊20C22, (C2×SD16)⋊32C22, C4⋊1D4.81C22, C4.10D4⋊5C22, 2-Sylow(CSO+(4,3)), SmallGroup(128,2023)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8⋊6D4
G = < a,b,c,d | a8=b2=c4=d2=1, bab=a-1, ac=ca, dad=a3, cbc-1=a6b, bd=db, dcd=c-1 >
Subgroups: 492 in 230 conjugacy classes, 92 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C42, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), D8, D8, SD16, SD16, Q16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4×C8, C4.D4, C4.10D4, C4≀C2, C8.C4, C4⋊1D4, C4⋊Q8, C8○D4, C2×SD16, C2×SD16, C4○D8, C4○D8, C8⋊C22, C8⋊C22, C8.C22, C8.C22, 2+ 1+4, 2- 1+4, C8○D8, D4⋊4D4, D4.10D4, D4.3D4, C8⋊5D4, D4○SD16, D8⋊6D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, D42, D8⋊6D4
Character table of D8⋊6D4
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | |
size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 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 | -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 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | linear of order 2 |
ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 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 |
ρ6 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 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 |
ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -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 |
ρ8 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -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 |
ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 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 |
ρ10 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 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 |
ρ11 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -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 |
ρ12 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -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 |
ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 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 |
ρ14 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 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 |
ρ15 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -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 |
ρ16 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -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 |
ρ17 | 2 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ18 | 2 | 2 | -2 | 0 | -2 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ19 | 2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ20 | 2 | 2 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ21 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ22 | 2 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ23 | 2 | 2 | -2 | 0 | 2 | 0 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ24 | 2 | 2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ25 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2√-2 | 2√-2 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2√-2 | -2√-2 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2√-2 | -2√-2 | -2√-2 | 2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2√-2 | 2√-2 | 2√-2 | -2√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 16)(8 15)
(1 5)(2 6)(3 7)(4 8)(9 11 13 15)(10 12 14 16)
(1 5)(2 8)(4 6)(9 11)(10 14)(13 15)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,16)(8,15), (1,5)(2,6)(3,7)(4,8)(9,11,13,15)(10,12,14,16), (1,5)(2,8)(4,6)(9,11)(10,14)(13,15)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,16)(8,15), (1,5)(2,6)(3,7)(4,8)(9,11,13,15)(10,12,14,16), (1,5)(2,8)(4,6)(9,11)(10,14)(13,15) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,16),(8,15)], [(1,5),(2,6),(3,7),(4,8),(9,11,13,15),(10,12,14,16)], [(1,5),(2,8),(4,6),(9,11),(10,14),(13,15)]])
G:=TransitiveGroup(16,328);
Matrix representation of D8⋊6D4 ►in GL4(𝔽3) generated by
1 | 0 | 1 | 0 |
2 | 0 | 1 | 1 |
0 | 1 | 2 | 0 |
2 | 1 | 2 | 0 |
0 | 0 | 1 | 2 |
2 | 0 | 1 | 1 |
2 | 2 | 2 | 1 |
1 | 2 | 2 | 1 |
0 | 1 | 2 | 1 |
0 | 1 | 0 | 2 |
1 | 2 | 2 | 0 |
2 | 1 | 2 | 2 |
2 | 2 | 1 | 0 |
1 | 2 | 0 | 2 |
1 | 1 | 1 | 2 |
1 | 2 | 1 | 1 |
G:=sub<GL(4,GF(3))| [1,2,0,2,0,0,1,1,1,1,2,2,0,1,0,0],[0,2,2,1,0,0,2,2,1,1,2,2,2,1,1,1],[0,0,1,2,1,1,2,1,2,0,2,2,1,2,0,2],[2,1,1,1,2,2,1,2,1,0,1,1,0,2,2,1] >;
D8⋊6D4 in GAP, Magma, Sage, TeX
D_8\rtimes_6D_4
% in TeX
G:=Group("D8:6D4");
// GroupNames label
G:=SmallGroup(128,2023);
// by ID
G=gap.SmallGroup(128,2023);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,723,346,2804,1411,375,172,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^3,c*b*c^-1=a^6*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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