p-group, metacyclic, nilpotent (class 3), monomial
Aliases: C8⋊2C8, C4.15SD16, C4.4M4(2), C42.64C22, C4⋊C8.4C2, C4.6(C2×C8), C2.3(C4⋊C8), (C2×C4).8Q8, (C2×C8).11C4, (C4×C8).11C2, (C2×C4).138D4, C2.1(C4.Q8), C2.1(C8.C4), C22.12(C4⋊C4), (C2×C4).59(C2×C4), SmallGroup(64,15)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊2C8
G = < a,b | a8=b8=1, bab-1=a3 >
Character table of C8⋊2C8
| 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 | 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 |
| ρ19 | 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) |
| ρ20 | 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) |
| ρ21 | 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 |
| ρ22 | 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 |
| ρ23 | 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 | complex lifted from SD16 |
| ρ24 | 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 | complex lifted from SD16 |
| ρ25 | 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 | complex lifted from SD16 |
| 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 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ28 | 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 |
►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 22 55 60 42 25 38 10)(2 17 56 63 43 28 39 13)(3 20 49 58 44 31 40 16)(4 23 50 61 45 26 33 11)(5 18 51 64 46 29 34 14)(6 21 52 59 47 32 35 9)(7 24 53 62 48 27 36 12)(8 19 54 57 41 30 37 15)
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), (1,22,55,60,42,25,38,10)(2,17,56,63,43,28,39,13)(3,20,49,58,44,31,40,16)(4,23,50,61,45,26,33,11)(5,18,51,64,46,29,34,14)(6,21,52,59,47,32,35,9)(7,24,53,62,48,27,36,12)(8,19,54,57,41,30,37,15)>;
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), (1,22,55,60,42,25,38,10)(2,17,56,63,43,28,39,13)(3,20,49,58,44,31,40,16)(4,23,50,61,45,26,33,11)(5,18,51,64,46,29,34,14)(6,21,52,59,47,32,35,9)(7,24,53,62,48,27,36,12)(8,19,54,57,41,30,37,15) );
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)], [(1,22,55,60,42,25,38,10),(2,17,56,63,43,28,39,13),(3,20,49,58,44,31,40,16),(4,23,50,61,45,26,33,11),(5,18,51,64,46,29,34,14),(6,21,52,59,47,32,35,9),(7,24,53,62,48,27,36,12),(8,19,54,57,41,30,37,15)]])
C8⋊2C8 is a maximal subgroup of
D8⋊C8 Q16⋊C8 C42.42Q8 M4(2)⋊1C8 C8⋊8M4(2) C42.43Q8 C8⋊1M4(2) C42.90D4 C42.Q8 C42.92D4 C42.21Q8 C8×SD16 Q16⋊5C8 D8⋊5C8 C8⋊12SD16 C8⋊15SD16 D4⋊2M4(2) Q8.M4(2) C8⋊9SD16 C8.M4(2) C8⋊3M4(2) C8⋊8D8 C8⋊14SD16 C8⋊11SD16 C8⋊8Q16 D4.2SD16 Q8.2SD16 D4.3SD16 Q8.3SD16 C8⋊2D8 C8⋊2SD16 C8.D8 C8⋊4SD16 C8⋊2Q16 C8.3Q16 C42.252C23 C42.253C23 C42.254C23 C42.255C23 C20.26M4(2)
C8p⋊C8: C16⋊1C8 C24⋊2C8 C40⋊6C8 C40⋊2C8 C56⋊2C8 ...
C4p.SD16: C8.30D8 C8.16Q16 C8.SD16 C8.8SD16 C12.39SD16 C20.39SD16 C28.39SD16 ...
C8⋊2C8 is a maximal quotient of
C42.385D4 C20.26M4(2)
C8p⋊C8: C16⋊1C8 C24⋊2C8 C40⋊6C8 C40⋊2C8 C56⋊2C8 ...
C4p.SD16: C8⋊2C16 C16.C8 C12.39SD16 C20.39SD16 C28.39SD16 ...
Matrix representation of C8⋊2C8 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 0 | 10 |
| 0 | 12 | 10 |
| 2 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 6 | 0 |
G:=sub<GL(3,GF(17))| [1,0,0,0,0,12,0,10,10],[2,0,0,0,0,6,0,12,0] >;
C8⋊2C8 in GAP, Magma, Sage, TeX
C_8\rtimes_2C_8
% in TeX
G:=Group("C8:2C8"); // GroupNames label
G:=SmallGroup(64,15);
// by ID
G=gap.SmallGroup(64,15);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,31,362,86,117]);
// Polycyclic
G:=Group<a,b|a^8=b^8=1,b*a*b^-1=a^3>;
// generators/relations
Export
Subgroup lattice of C8⋊2C8 in TeX
Character table of C8⋊2C8 in TeX