p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.1D4, C23.18D4, C4.Q8⋊5C2, (C2×Q16)⋊7C2, (C2×C4).32D4, C4.58(C2×D4), C22⋊Q8.4C2, Q8⋊C4⋊18C2, C4.12(C4○D4), C4⋊C4.10C22, (C2×C4).98C23, (C2×C8).53C22, C22.94(C2×D4), C2.22(C4⋊D4), (C2×M4(2)).3C2, (C2×Q8).14C22, C2.13(C8.C22), (C22×C4).50C22, SmallGroup(64,151)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.D4
G = < a,b,c | a8=b4=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a4b-1 >
Character table of C8.D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | complex lifted from C4○D4 |
| ρ15 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ16 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►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 14 19 32)(2 9 20 27)(3 12 21 30)(4 15 22 25)(5 10 23 28)(6 13 24 31)(7 16 17 26)(8 11 18 29)
(1 28 5 32)(2 27 6 31)(3 26 7 30)(4 25 8 29)(9 24 13 20)(10 23 14 19)(11 22 15 18)(12 21 16 17)
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), (1,14,19,32)(2,9,20,27)(3,12,21,30)(4,15,22,25)(5,10,23,28)(6,13,24,31)(7,16,17,26)(8,11,18,29), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,24,13,20)(10,23,14,19)(11,22,15,18)(12,21,16,17)>;
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), (1,14,19,32)(2,9,20,27)(3,12,21,30)(4,15,22,25)(5,10,23,28)(6,13,24,31)(7,16,17,26)(8,11,18,29), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,24,13,20)(10,23,14,19)(11,22,15,18)(12,21,16,17) );
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)], [(1,14,19,32),(2,9,20,27),(3,12,21,30),(4,15,22,25),(5,10,23,28),(6,13,24,31),(7,16,17,26),(8,11,18,29)], [(1,28,5,32),(2,27,6,31),(3,26,7,30),(4,25,8,29),(9,24,13,20),(10,23,14,19),(11,22,15,18),(12,21,16,17)]])
C8.D4 is a maximal subgroup of
C23.2SD16 C23.10SD16 C24.110D4 M4(2)⋊15D4 C8.D4⋊C2 (C2×C8)⋊13D4 M4(2)⋊17D4 C42.260D4 C42.262D4 C24.128D4 C24.129D4 C24.130D4 C4.162+ 1+4 C4.172+ 1+4 C4.192+ 1+4 C42.300D4 C42.303D4 C42.304D4
C23.D4p: C23.2D8 D8.D4 C24.4D4 C40.4D4 C56.4D4 ...
C4⋊C4.D2p: C42.385C23 C42.389C23 C42.390C23 C42.25C23 C42.28C23 C42.30C23 D8⋊10D4 Q16⋊9D4 ...
(C2p×Q16)⋊C2: C42.256D4 C24.36D4 C40.36D4 C56.36D4 ...
C8.D4 is a maximal quotient of
C8.D4p: C8.D8 C8.2D12 C8.2D20 C8.2D28 ...
C23.D4p: C23.12D8 C24.4D4 C40.4D4 C56.4D4 ...
C4⋊C4.D2p: C8.SD16 C8.8SD16 C8.3Q16 C42.254C23 C42.255C23 C23⋊2Q16 C24.85D4 (C2×Q8).8Q8 ...
(C2×C8).D2p: C24.67D4 C24.75D4 C2.(C8⋊D4) C8⋊(C4⋊C4) (C2×Q16)⋊10C4 (C2×C4)⋊3Q16 (C2×C4).26D8 C24.36D4 ...
Matrix representation of C8.D4 ►in GL6(𝔽17)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 11 | 15 | 0 |
| 0 | 0 | 11 | 7 | 0 | 15 |
| 0 | 0 | 0 | 1 | 10 | 6 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 | 7 | 10 |
| 0 | 0 | 12 | 12 | 10 | 10 |
| 0 | 0 | 7 | 6 | 3 | 5 |
| 0 | 0 | 6 | 5 | 14 | 5 |
| 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 | 7 | 10 |
| 0 | 0 | 12 | 12 | 10 | 10 |
| 0 | 0 | 2 | 11 | 3 | 5 |
| 0 | 0 | 11 | 10 | 14 | 5 |
G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,7,11,0,0,0,0,11,7,1,0,0,0,15,0,10,6,0,0,0,15,6,10],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,14,12,7,6,0,0,3,12,6,5,0,0,7,10,3,14,0,0,10,10,5,5],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,14,12,2,11,0,0,3,12,11,10,0,0,7,10,3,14,0,0,10,10,5,5] >;
C8.D4 in GAP, Magma, Sage, TeX
C_8.D_4
% in TeX
G:=Group("C8.D4"); // GroupNames label
G:=SmallGroup(64,151);
// by ID
G=gap.SmallGroup(64,151);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,247,362,332,963,117]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=1,c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations
Export
Subgroup lattice of C8.D4 in TeX
Character table of C8.D4 in TeX