p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C8.2D4, C42.32C22, C4⋊Q8⋊8C2, C8⋊C4⋊5C2, C4.6(C2×D4), (C2×Q16)⋊9C2, (C2×C4).45D4, (C2×C8).21C22, (C2×SD16).2C2, C4.4D4.7C2, C2.10(C4⋊1D4), (C2×C4).122C23, (C2×D4).32C22, C22.118(C2×D4), (C2×Q8).28C22, C2.22(C8.C22), SmallGroup(64,178)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.2D4
G = < a,b,c | a8=c2=1, b4=a4, bab-1=a5, cac=a3, cbc=b3 >
Subgroups: 113 in 62 conjugacy classes, 29 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C8.2D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4⋊1D4, C8.C22, C8.2D4
Character table of C8.2D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 4 | 4 | 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 | 0 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | orthogonal lifted from 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 15 30 22 5 11 26 18)(2 12 31 19 6 16 27 23)(3 9 32 24 7 13 28 20)(4 14 25 21 8 10 29 17)
(1 30)(2 25)(3 28)(4 31)(5 26)(6 29)(7 32)(8 27)(10 12)(11 15)(14 16)(17 23)(19 21)(20 24)
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), (1,15,30,22,5,11,26,18)(2,12,31,19,6,16,27,23)(3,9,32,24,7,13,28,20)(4,14,25,21,8,10,29,17), (1,30)(2,25)(3,28)(4,31)(5,26)(6,29)(7,32)(8,27)(10,12)(11,15)(14,16)(17,23)(19,21)(20,24)>;
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), (1,15,30,22,5,11,26,18)(2,12,31,19,6,16,27,23)(3,9,32,24,7,13,28,20)(4,14,25,21,8,10,29,17), (1,30)(2,25)(3,28)(4,31)(5,26)(6,29)(7,32)(8,27)(10,12)(11,15)(14,16)(17,23)(19,21)(20,24) );
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)], [(1,15,30,22,5,11,26,18),(2,12,31,19,6,16,27,23),(3,9,32,24,7,13,28,20),(4,14,25,21,8,10,29,17)], [(1,30),(2,25),(3,28),(4,31),(5,26),(6,29),(7,32),(8,27),(10,12),(11,15),(14,16),(17,23),(19,21),(20,24)]])
C8.2D4 is a maximal subgroup of
(C2×D4).D4 C42.5C23 C42.6C23 C42.7C23 M4(2)⋊10D4 C42.390C23 C42.409C23 C42.73C23 C42.75C23 C42.532C23
C42.D2p: C42.247D4 M4(2)⋊8D4 M4(2)⋊9D4 C42.256D4 C42.258D4 C42.273D4 C42.274D4 C42.276D4 ...
(C2×C8).D2p: M4(2).20D4 C24.31D4 C24.37D4 C40.31D4 C40.37D4 C56.31D4 C56.37D4 ...
D4.pD4⋊C2: C42.408C23 C42.411C23 SD16⋊6D4 D8⋊10D4 SD16⋊8D4 Q16⋊9D4 D8.13D4 C42.531C23 ...
C8.2D4 is a maximal quotient of
C42.D2p: C42.110D4 C42.111D4 C8.D12 C42.65D6 C42.80D6 C8.D20 C42.65D10 C42.80D10 ...
(C2×C8).D2p: C8⋊5SD16 C8.9SD16 C42.665C23 C42.666C23 C42.667C23 C8.2D8 C8⋊3Q16 C42.26Q8 ...
Matrix representation of C8.2D4 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,4,0,0,0,0,0,0,4,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;
C8.2D4 in GAP, Magma, Sage, TeX
C_8._2D_4
% in TeX
G:=Group("C8.2D4"); // GroupNames label
G:=SmallGroup(64,178);
// by ID
G=gap.SmallGroup(64,178);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,55,362,332,86,963,117]);
// Polycyclic
G:=Group<a,b,c|a^8=c^2=1,b^4=a^4,b*a*b^-1=a^5,c*a*c=a^3,c*b*c=b^3>;
// generators/relations
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