p-group, metacyclic, nilpotent (class 3), monomial
Aliases: C8⋊1C8, C4.17D8, C4.9Q16, C4.5M4(2), C42.65C22, C4⋊C8.5C2, C4.7(C2×C8), (C2×C8).8C4, (C4×C8).5C2, C2.4(C4⋊C8), (C2×C4).9Q8, (C2×C4).139D4, C2.1(C2.D8), C2.2(C8.C4), C22.13(C4⋊C4), (C2×C4).60(C2×C4), SmallGroup(64,16)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊1C8
G = < a,b | a8=b8=1, bab-1=a-1 >
Character table of C8⋊1C8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 8M | 8N | 8O | 8P | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 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 | -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 | -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 | -i | i | -i | i | i | -i | i | -i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -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 | 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 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ9 | 1 | -1 | 1 | -1 | i | -i | i | -i | i | -i | 1 | -1 | -1 | 1 | i | 1 | -1 | -i | i | -i | ζ87 | ζ83 | ζ83 | ζ87 | ζ85 | ζ8 | ζ8 | ζ85 | linear of order 8 |
| ρ10 | 1 | -1 | 1 | -1 | i | -i | i | -i | i | -i | 1 | -1 | -1 | 1 | i | 1 | -1 | -i | i | -i | ζ83 | ζ87 | ζ87 | ζ83 | ζ8 | ζ85 | ζ85 | ζ8 | linear of order 8 |
| ρ11 | 1 | -1 | 1 | -1 | i | -i | i | -i | i | -i | 1 | -1 | 1 | -1 | -i | -1 | 1 | i | -i | i | ζ83 | ζ83 | ζ87 | ζ87 | ζ8 | ζ8 | ζ85 | ζ85 | linear of order 8 |
| ρ12 | 1 | -1 | 1 | -1 | i | -i | i | -i | i | -i | 1 | -1 | 1 | -1 | -i | -1 | 1 | i | -i | i | ζ87 | ζ87 | ζ83 | ζ83 | ζ85 | ζ85 | ζ8 | ζ8 | linear of order 8 |
| ρ13 | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | i | 1 | -1 | -1 | 1 | -i | 1 | -1 | i | -i | i | ζ8 | ζ85 | ζ85 | ζ8 | ζ83 | ζ87 | ζ87 | ζ83 | linear of order 8 |
| ρ14 | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | i | 1 | -1 | 1 | -1 | i | -1 | 1 | -i | i | -i | ζ85 | ζ85 | ζ8 | ζ8 | ζ87 | ζ87 | ζ83 | ζ83 | linear of order 8 |
| ρ15 | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | i | 1 | -1 | -1 | 1 | -i | 1 | -1 | i | -i | i | ζ85 | ζ8 | ζ8 | ζ85 | ζ87 | ζ83 | ζ83 | ζ87 | linear of order 8 |
| ρ16 | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | i | 1 | -1 | 1 | -1 | i | -1 | 1 | -i | i | -i | ζ8 | ζ8 | ζ85 | ζ85 | ζ83 | ζ83 | ζ87 | ζ87 | linear of order 8 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | -√2 | -√2 | √2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ19 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | √2 | √2 | -√2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ20 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | √2 | -√2 | √2 | -√2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ21 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ22 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | -√2 | √2 | -√2 | √2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ23 | 2 | -2 | 2 | -2 | -2i | 2i | -2i | 2i | 2i | -2i | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ24 | 2 | -2 | -2 | 2 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | √2 | √2 | -√-2 | -√2 | -√2 | √-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.C4 |
| ρ25 | 2 | -2 | -2 | 2 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | -√2 | -√2 | -√-2 | √2 | √2 | √-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.C4 |
| ρ26 | 2 | -2 | -2 | 2 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | √2 | √2 | √-2 | -√2 | -√2 | -√-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.C4 |
| ρ27 | 2 | -2 | -2 | 2 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | -√2 | -√2 | √-2 | √2 | √2 | -√-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C8.C4 |
| ρ28 | 2 | -2 | 2 | -2 | 2i | -2i | 2i | -2i | -2i | 2i | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
►Regular action on 64 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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 19 50 16 42 31 34 64)(2 18 51 15 43 30 35 63)(3 17 52 14 44 29 36 62)(4 24 53 13 45 28 37 61)(5 23 54 12 46 27 38 60)(6 22 55 11 47 26 39 59)(7 21 56 10 48 25 40 58)(8 20 49 9 41 32 33 57)
G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,19,50,16,42,31,34,64)(2,18,51,15,43,30,35,63)(3,17,52,14,44,29,36,62)(4,24,53,13,45,28,37,61)(5,23,54,12,46,27,38,60)(6,22,55,11,47,26,39,59)(7,21,56,10,48,25,40,58)(8,20,49,9,41,32,33,57)>;
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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,19,50,16,42,31,34,64)(2,18,51,15,43,30,35,63)(3,17,52,14,44,29,36,62)(4,24,53,13,45,28,37,61)(5,23,54,12,46,27,38,60)(6,22,55,11,47,26,39,59)(7,21,56,10,48,25,40,58)(8,20,49,9,41,32,33,57) );
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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,19,50,16,42,31,34,64),(2,18,51,15,43,30,35,63),(3,17,52,14,44,29,36,62),(4,24,53,13,45,28,37,61),(5,23,54,12,46,27,38,60),(6,22,55,11,47,26,39,59),(7,21,56,10,48,25,40,58),(8,20,49,9,41,32,33,57)]])
C8⋊1C8 is a maximal subgroup of
C4.16D16 Q16⋊1C8 C4.10D16 C4.6Q32 C42.42Q8 M4(2)⋊1C8 C8⋊7M4(2) C42.43Q8 C8⋊1M4(2) C42.91D4 C42.Q8 C42.92D4 C42.21Q8 C8×D8 C8×Q16 SD16⋊C8 C8⋊9D8 C8⋊9Q16 D4.M4(2) Q8⋊2M4(2) C8⋊6D8 C8⋊6Q16 C8⋊M4(2) C8⋊7D8 C8⋊13SD16 Q8⋊1Q16 C8⋊10SD16 C8⋊7Q16 D4.1Q16 Q8.1Q16 D4.2D8 Q8.2D8 D4.Q16 Q8.2Q16 C8⋊D8 C8⋊SD16 C8⋊3SD16 C8⋊Q16 C42.248C23 C42.249C23 C42.250C23 C42.251C23 Dic5.13D8
C8p⋊C8: C16⋊3C8 C16⋊4C8 C24⋊1C8 C40⋊5C8 C40⋊1C8 C56⋊1C8 ...
C4p.D8: C4.D16 C8.27D8 C8.28D8 C12.53D8 C20.53D8 C28.53D8 ...
C8⋊1C8 is a maximal quotient of
C42.385D4 C16⋊3C8 C16⋊4C8 C40⋊1C8 Dic5.13D8
C4p.D8: C8.36D8 C16.3C8 C24⋊1C8 C12.53D8 C40⋊5C8 C20.53D8 C56⋊1C8 C28.53D8 ...
Matrix representation of C8⋊1C8 ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 9 |
| 8 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(17))| [16,0,0,0,2,0,0,0,9],[8,0,0,0,0,1,0,1,0] >;
C8⋊1C8 in GAP, Magma, Sage, TeX
C_8\rtimes_1C_8
% in TeX
G:=Group("C8:1C8"); // GroupNames label
G:=SmallGroup(64,16);
// by ID
G=gap.SmallGroup(64,16);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,127,362,86,117]);
// Polycyclic
G:=Group<a,b|a^8=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of C8⋊1C8 in TeX
Character table of C8⋊1C8 in TeX