p-group, metabelian, nilpotent (class 4), monomial
Aliases: C8.17D4, C4.12D8, Q16.2C4, M5(2).3C2, C22.4SD16, C8.3(C2×C4), (C2×C4).12D4, (C2×Q16).5C2, C8.C4.1C2, C4.6(C22⋊C4), (C2×C8).11C22, C2.11(D4⋊C4), SmallGroup(64,43)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.17D4
G = < a,b,c | a8=1, b4=a4, c2=bab-1=a-1, ac=ca, cbc-1=a-1b3 >
Character table of C8.17D4
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 16A | 16B | 16C | 16D | |
| size | 1 | 1 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 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 | i | -i | i | -i | i | -i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | i | -i | -i | i | -i | i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -i | i | -i | i | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | i | i | -i | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ13 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ14 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 32 points
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 32 30 28 26 24 22 20)
(1 22 13 26 9 30 5 18)(2 25 6 21 10 17 14 29)(3 20 15 24 11 28 7 32)(4 23 8 19 12 31 16 27)
(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)
G:=sub<Sym(32)| (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20), (1,22,13,26,9,30,5,18)(2,25,6,21,10,17,14,29)(3,20,15,24,11,28,7,32)(4,23,8,19,12,31,16,27), (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)>;
G:=Group( (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20), (1,22,13,26,9,30,5,18)(2,25,6,21,10,17,14,29)(3,20,15,24,11,28,7,32)(4,23,8,19,12,31,16,27), (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) );
G=PermutationGroup([[(1,15,13,11,9,7,5,3),(2,16,14,12,10,8,6,4),(17,31,29,27,25,23,21,19),(18,32,30,28,26,24,22,20)], [(1,22,13,26,9,30,5,18),(2,25,6,21,10,17,14,29),(3,20,15,24,11,28,7,32),(4,23,8,19,12,31,16,27)], [(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)]])
C8.17D4 is a maximal subgroup of
C23.21SD16 Q32⋊C4 Q16.D4 D8.12D4 D4.4D8 D4.5D8 C23.10SD16 Dic20.C4
C4p.D8: C8.3D8 C8.5D8 C24.8D4 C12.4D8 Q16.Dic3 C40.8D4 C20.4D8 Q16.Dic5 ...
C8.17D4 is a maximal quotient of
Q16⋊C8 C23.13SD16 C4.10D16 C8.2C42 Dic20.C4
C4p.D8: C8.27D8 C24.8D4 C12.4D8 Q16.Dic3 C40.8D4 C20.4D8 Q16.Dic5 C56.8D4 ...
Matrix representation of C8.17D4 ►in GL4(𝔽7) generated by
| 2 | 0 | 5 | 1 |
| 1 | 2 | 2 | 1 |
| 1 | 6 | 6 | 5 |
| 5 | 5 | 1 | 5 |
| 4 | 0 | 5 | 5 |
| 6 | 2 | 2 | 5 |
| 1 | 1 | 0 | 5 |
| 0 | 3 | 4 | 1 |
| 0 | 2 | 5 | 5 |
| 2 | 2 | 0 | 1 |
| 6 | 1 | 4 | 4 |
| 2 | 1 | 2 | 1 |
G:=sub<GL(4,GF(7))| [2,1,1,5,0,2,6,5,5,2,6,1,1,1,5,5],[4,6,1,0,0,2,1,3,5,2,0,4,5,5,5,1],[0,2,6,2,2,2,1,1,5,0,4,2,5,1,4,1] >;
C8.17D4 in GAP, Magma, Sage, TeX
C_8._{17}D_4 % in TeX
G:=Group("C8.17D4"); // GroupNames label
G:=SmallGroup(64,43);
// by ID
G=gap.SmallGroup(64,43);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,476,86,489,117,1444,730,88]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^4=a^4,c^2=b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^3>;
// generators/relations
Export
Subgroup lattice of C8.17D4 in TeX
Character table of C8.17D4 in TeX