p-group, metabelian, nilpotent (class 2), monomial
Aliases: Q8:6D4, C23.17C23, C42.46C22, C22.44C24, C2.142+ 1+4, Q8o2(C4:C4), (C4xD4):18C2, C4:3(C4oD4), C4:1D4:8C2, (C4xQ8):13C2, C4.40(C2xD4), C4:D4:14C2, C4:C4.82C22, (C2xC4).55C23, C2.22(C22xD4), (C2xD4).69C22, (C2xQ8).74C22, C22:C4.21C22, (C22xC4).71C22, C4:C4o(C2xQ8), (C2xC4oD4):9C2, C2.23(C2xC4oD4), SmallGroup(64,231)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8:6D4
G = < a,b,c,d | a4=c4=d2=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 253 in 156 conjugacy classes, 83 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C42, C22:C4, C4:C4, C4:C4, C22xC4, C2xD4, C2xQ8, C4oD4, C4xD4, C4xQ8, C4:D4, C4:1D4, C2xC4oD4, Q8:6D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C22xD4, C2xC4oD4, 2+ 1+4, Q8:6D4
Character table of Q8:6D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | -2 | 0 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 2 | 0 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | -2 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2i | 0 | 0 | -2i | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2i | 0 | 0 | 2i | 0 | -2i | 0 | 0 | -2i | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2i | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 2i | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -2i | 0 | 0 | 2i | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
►On 32 points
(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)
(1 16 3 14)(2 15 4 13)(5 24 7 22)(6 23 8 21)(9 19 11 17)(10 18 12 20)(25 29 27 31)(26 32 28 30)
(1 31 12 8)(2 30 9 7)(3 29 10 6)(4 32 11 5)(13 26 17 22)(14 25 18 21)(15 28 19 24)(16 27 20 23)
(1 28)(2 25)(3 26)(4 27)(5 20)(6 17)(7 18)(8 19)(9 21)(10 22)(11 23)(12 24)(13 29)(14 30)(15 31)(16 32)
G:=sub<Sym(32)| (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), (1,16,3,14)(2,15,4,13)(5,24,7,22)(6,23,8,21)(9,19,11,17)(10,18,12,20)(25,29,27,31)(26,32,28,30), (1,31,12,8)(2,30,9,7)(3,29,10,6)(4,32,11,5)(13,26,17,22)(14,25,18,21)(15,28,19,24)(16,27,20,23), (1,28)(2,25)(3,26)(4,27)(5,20)(6,17)(7,18)(8,19)(9,21)(10,22)(11,23)(12,24)(13,29)(14,30)(15,31)(16,32)>;
G:=Group( (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), (1,16,3,14)(2,15,4,13)(5,24,7,22)(6,23,8,21)(9,19,11,17)(10,18,12,20)(25,29,27,31)(26,32,28,30), (1,31,12,8)(2,30,9,7)(3,29,10,6)(4,32,11,5)(13,26,17,22)(14,25,18,21)(15,28,19,24)(16,27,20,23), (1,28)(2,25)(3,26)(4,27)(5,20)(6,17)(7,18)(8,19)(9,21)(10,22)(11,23)(12,24)(13,29)(14,30)(15,31)(16,32) );
G=PermutationGroup([[(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)], [(1,16,3,14),(2,15,4,13),(5,24,7,22),(6,23,8,21),(9,19,11,17),(10,18,12,20),(25,29,27,31),(26,32,28,30)], [(1,31,12,8),(2,30,9,7),(3,29,10,6),(4,32,11,5),(13,26,17,22),(14,25,18,21),(15,28,19,24),(16,27,20,23)], [(1,28),(2,25),(3,26),(4,27),(5,20),(6,17),(7,18),(8,19),(9,21),(10,22),(11,23),(12,24),(13,29),(14,30),(15,31),(16,32)]])
Q8:6D4 is a maximal subgroup of
Q8:2SD16 Q8.D8 Q8:3SD16 Q8:8SD16 C42.507C23 C42.509C23 D4xC4oD4 C22.69C25 C22.76C25 C22.77C25 C4:2- 1+4 C22.87C25 C22.111C25 C22.113C25 SL2(F3):6D4
Q8:D4p: Q8:D8 Q8:4D8 Q8:5D8 Q8:7D12 Q8:6D20 Q8:6D28 ...
C2p.2+ 1+4: SD16:7D4 Q16:10D4 SD16:1D4 Q16:5D4 SD16:11D4 Q16:13D4 C42.469C23 C42.470C23 ...
Q8:6D4 is a maximal quotient of
C23.223C24 Q8xC4:C4 C23.236C24 C24.217C23 C24.219C23 C24.244C23 C24.249C23 C23.323C24 C24.259C23 C23.328C24 C24.263C23 C24.268C23 C23.345C24 C23.348C24 C23.354C24 C23.356C24 C23.367C24 C24.290C23 C23.379C24 C23.390C24 C23.412C24 C42.166D4 C42:20D4 C42.171D4 C42.35Q8 C24.327C23 C23.573C24 C24.389C23 C24.401C23 C23.605C24 C24.413C23 C23.621C24 C23.627C24 C23.630C24 C23.632C24 C23.633C24
Q8:D4p: Q8:5D8 Q8:7D12 Q8:6D20 Q8:6D28 ...
C2p.2+ 1+4: Q8:9SD16 C42.527C23 C42.528C23 Q8:6Q16 C42.530C23 C42.72C23 C42.73C23 C42.74C23 ...
Matrix representation of Q8:6D4 ►in GL4(F5) generated by
| 3 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 2 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 4 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(5))| [3,0,0,0,0,2,0,0,0,0,4,0,0,0,0,4],[0,4,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[0,2,0,0,2,0,0,0,0,0,1,4,0,0,2,4],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,3,1] >;
Q8:6D4 in GAP, Magma, Sage, TeX
Q_8\rtimes_6D_4
% in TeX
G:=Group("Q8:6D4"); // GroupNames label
G:=SmallGroup(64,231);
// by ID
G=gap.SmallGroup(64,231);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,103,650,86,297,69]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^4=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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