p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.9D8, C4.11SD16, C42.3C22, C4⋊C8⋊2C2, (C2×D4).2C4, C4⋊1D4.1C2, (C2×C4).108D4, C2.4(D4⋊C4), C2.4(C4.D4), C22.39(C22⋊C4), (C2×C4).12(C2×C4), SmallGroup(64,12)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.D8
G = < a,b,c | a4=b8=1, c2=a, bab-1=a-1, ac=ca, cbc-1=ab-1 >
Character table of C4.D8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | -i | i | i | -i | i | -i | i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | i | i | -i | -i | i | -i | i | -i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | i | -i | i | -i | i | i | -i | -i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -i | i | -i | i | -i | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | √2 | √2 | 0 | 0 | -√2 | -√2 | 0 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | √2 | 0 | 0 | √2 | -√2 | 0 | 0 | -√2 | orthogonal lifted from D8 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | -√2 | -√2 | 0 | 0 | √2 | √2 | 0 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -√2 | 0 | 0 | -√2 | √2 | 0 | 0 | √2 | orthogonal lifted from D8 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | -√-2 | 0 | 0 | √-2 | √-2 | 0 | 0 | -√-2 | complex lifted from SD16 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | √-2 | -√-2 | 0 | 0 | √-2 | -√-2 | 0 | complex lifted from SD16 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -√-2 | √-2 | 0 | 0 | -√-2 | √-2 | 0 | complex lifted from SD16 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | √-2 | 0 | 0 | -√-2 | -√-2 | 0 | 0 | √-2 | complex lifted from SD16 |
| ρ19 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
►On 32 points
(1 24 29 9)(2 10 30 17)(3 18 31 11)(4 12 32 19)(5 20 25 13)(6 14 26 21)(7 22 27 15)(8 16 28 23)
(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 24 28 29 23 9 8)(2 7 10 22 30 27 17 15)(3 14 18 26 31 21 11 6)(4 5 12 20 32 25 19 13)
G:=sub<Sym(32)| (1,24,29,9)(2,10,30,17)(3,18,31,11)(4,12,32,19)(5,20,25,13)(6,14,26,21)(7,22,27,15)(8,16,28,23), (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), (1,16,24,28,29,23,9,8)(2,7,10,22,30,27,17,15)(3,14,18,26,31,21,11,6)(4,5,12,20,32,25,19,13)>;
G:=Group( (1,24,29,9)(2,10,30,17)(3,18,31,11)(4,12,32,19)(5,20,25,13)(6,14,26,21)(7,22,27,15)(8,16,28,23), (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), (1,16,24,28,29,23,9,8)(2,7,10,22,30,27,17,15)(3,14,18,26,31,21,11,6)(4,5,12,20,32,25,19,13) );
G=PermutationGroup([[(1,24,29,9),(2,10,30,17),(3,18,31,11),(4,12,32,19),(5,20,25,13),(6,14,26,21),(7,22,27,15),(8,16,28,23)], [(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)], [(1,16,24,28,29,23,9,8),(2,7,10,22,30,27,17,15),(3,14,18,26,31,21,11,6),(4,5,12,20,32,25,19,13)]])
C4.D8 is a maximal subgroup of
D4⋊D8 C42.181C23 Q8⋊D8 D4⋊2SD16 C42.191C23 Q8⋊2SD16 D4.D8 C42.201C23 Q8.D8 Q8⋊3SD16 C8⋊8D8 C8⋊14SD16 C8⋊7D8 C8⋊13SD16 D4.2SD16 Q8.2SD16 D4.2D8 Q8.2D8 C8⋊D8 C8⋊SD16 C8⋊2D8 C8⋊2SD16 C42.248C23 C42.249C23 C42.252C23 C42.253C23 Dic5.SD16
C42.D2p: C42.D4 C42.409D4 C42.411D4 C42.413D4 C42.78D4 C42.80D4 C42.417D4 C42.82D4 ...
C4.D8 is a maximal quotient of
(C2×C4).98D8 C42.8Q8 Dic5.SD16
C4p.D8: C8.24D8 C8.25D8 C8.29D8 C8.30D8 C4.D16 C8.27D8 C4.D24 C12.9D8 ...
Matrix representation of C4.D8 ►in GL4(𝔽17) generated by
| 16 | 15 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 7 | 0 | 0 |
| 5 | 10 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 14 | 0 |
| 7 | 7 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 14 | 11 |
G:=sub<GL(4,GF(17))| [16,1,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[7,5,0,0,7,10,0,0,0,0,6,14,0,0,6,0],[7,5,0,0,7,0,0,0,0,0,6,14,0,0,6,11] >;
C4.D8 in GAP, Magma, Sage, TeX
C_4.D_8
% in TeX
G:=Group("C4.D8"); // GroupNames label
G:=SmallGroup(64,12);
// by ID
G=gap.SmallGroup(64,12);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,362,332,158,681,165]);
// Polycyclic
G:=Group<a,b,c|a^4=b^8=1,c^2=a,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations
Export
Subgroup lattice of C4.D8 in TeX
Character table of C4.D8 in TeX