p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8⋊C8, C4.8Q16, C4.14SD16, C4.2M4(2), C42.63C22, C2.2C4≀C2, C4⋊C4.4C4, C4⋊C8.1C2, C4.2(C2×C8), (C4×C8).1C2, (C2×Q8).4C4, (C4×Q8).1C2, (C2×C4).137D4, C2.6(C22⋊C8), C2.1(Q8⋊C4), C22.22(C22⋊C4), (C2×C4).38(C2×C4), SmallGroup(64,7)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8⋊C8
G = < a,b,c | a4=c8=1, b2=a2, bab-1=cac-1=a-1, cbc-1=a-1b >
Character table of Q8⋊C8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | -i | i | i | i | 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 | i | -i | -i | -i | -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 | i | -i | -i | -i | -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 | -i | i | i | i | i | -i | -i | -i | -i | -i | i | i | linear of order 4 |
| ρ9 | 1 | -1 | 1 | -1 | -i | i | -i | i | i | 1 | -1 | -i | -1 | i | 1 | -i | ζ87 | ζ85 | ζ8 | ζ85 | ζ8 | ζ87 | ζ83 | ζ83 | ζ87 | ζ83 | ζ8 | ζ85 | linear of order 8 |
| ρ10 | 1 | -1 | 1 | -1 | -i | i | -i | i | i | 1 | -1 | -i | -1 | i | 1 | -i | ζ83 | ζ8 | ζ85 | ζ8 | ζ85 | ζ83 | ζ87 | ζ87 | ζ83 | ζ87 | ζ85 | ζ8 | linear of order 8 |
| ρ11 | 1 | -1 | 1 | -1 | -i | i | -i | i | i | 1 | -1 | -i | 1 | -i | -1 | i | ζ83 | ζ8 | ζ85 | ζ8 | ζ85 | ζ83 | ζ87 | ζ87 | ζ87 | ζ83 | ζ8 | ζ85 | linear of order 8 |
| ρ12 | 1 | -1 | 1 | -1 | -i | i | -i | i | i | 1 | -1 | -i | 1 | -i | -1 | i | ζ87 | ζ85 | ζ8 | ζ85 | ζ8 | ζ87 | ζ83 | ζ83 | ζ83 | ζ87 | ζ85 | ζ8 | linear of order 8 |
| ρ13 | 1 | -1 | 1 | -1 | i | -i | i | -i | -i | 1 | -1 | i | 1 | i | -1 | -i | ζ85 | ζ87 | ζ83 | ζ87 | ζ83 | ζ85 | ζ8 | ζ8 | ζ8 | ζ85 | ζ87 | ζ83 | linear of order 8 |
| ρ14 | 1 | -1 | 1 | -1 | i | -i | i | -i | -i | 1 | -1 | i | -1 | -i | 1 | i | ζ85 | ζ87 | ζ83 | ζ87 | ζ83 | ζ85 | ζ8 | ζ8 | ζ85 | ζ8 | ζ83 | ζ87 | linear of order 8 |
| ρ15 | 1 | -1 | 1 | -1 | i | -i | i | -i | -i | 1 | -1 | i | -1 | -i | 1 | i | ζ8 | ζ83 | ζ87 | ζ83 | ζ87 | ζ8 | ζ85 | ζ85 | ζ8 | ζ85 | ζ87 | ζ83 | linear of order 8 |
| ρ16 | 1 | -1 | 1 | -1 | i | -i | i | -i | -i | 1 | -1 | i | 1 | i | -1 | -i | ζ8 | ζ83 | ζ87 | ζ83 | ζ87 | ζ8 | ζ85 | ζ85 | ζ85 | ζ8 | ζ83 | 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 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | -√2 | √2 | -√2 | √2 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ20 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | √2 | -√2 | √2 | -√2 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ21 | 2 | 2 | -2 | -2 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1-i | -1-i | -1-i | 1+i | 1+i | -1+i | -1+i | 1-i | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ22 | 2 | 2 | -2 | -2 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+i | 1+i | 1+i | -1-i | -1-i | 1-i | 1-i | -1+i | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ23 | 2 | -2 | 2 | -2 | 2i | -2i | 2i | -2i | 2i | -2 | 2 | -2i | 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 | -2i | -2 | 2 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ25 | 2 | 2 | -2 | -2 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1+i | -1+i | -1+i | 1-i | 1-i | -1-i | -1-i | 1+i | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ26 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ27 | 2 | 2 | -2 | -2 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-i | 1-i | 1-i | -1+i | -1+i | 1+i | 1+i | -1-i | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
| ρ28 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
►Regular action on 64 points
(1 31 55 16)(2 9 56 32)(3 25 49 10)(4 11 50 26)(5 27 51 12)(6 13 52 28)(7 29 53 14)(8 15 54 30)(17 59 33 42)(18 43 34 60)(19 61 35 44)(20 45 36 62)(21 63 37 46)(22 47 38 64)(23 57 39 48)(24 41 40 58)
(1 59 55 42)(2 34 56 18)(3 61 49 44)(4 36 50 20)(5 63 51 46)(6 38 52 22)(7 57 53 48)(8 40 54 24)(9 43 32 60)(10 35 25 19)(11 45 26 62)(12 37 27 21)(13 47 28 64)(14 39 29 23)(15 41 30 58)(16 33 31 17)
(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)
G:=sub<Sym(64)| (1,31,55,16)(2,9,56,32)(3,25,49,10)(4,11,50,26)(5,27,51,12)(6,13,52,28)(7,29,53,14)(8,15,54,30)(17,59,33,42)(18,43,34,60)(19,61,35,44)(20,45,36,62)(21,63,37,46)(22,47,38,64)(23,57,39,48)(24,41,40,58), (1,59,55,42)(2,34,56,18)(3,61,49,44)(4,36,50,20)(5,63,51,46)(6,38,52,22)(7,57,53,48)(8,40,54,24)(9,43,32,60)(10,35,25,19)(11,45,26,62)(12,37,27,21)(13,47,28,64)(14,39,29,23)(15,41,30,58)(16,33,31,17), (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)>;
G:=Group( (1,31,55,16)(2,9,56,32)(3,25,49,10)(4,11,50,26)(5,27,51,12)(6,13,52,28)(7,29,53,14)(8,15,54,30)(17,59,33,42)(18,43,34,60)(19,61,35,44)(20,45,36,62)(21,63,37,46)(22,47,38,64)(23,57,39,48)(24,41,40,58), (1,59,55,42)(2,34,56,18)(3,61,49,44)(4,36,50,20)(5,63,51,46)(6,38,52,22)(7,57,53,48)(8,40,54,24)(9,43,32,60)(10,35,25,19)(11,45,26,62)(12,37,27,21)(13,47,28,64)(14,39,29,23)(15,41,30,58)(16,33,31,17), (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) );
G=PermutationGroup([[(1,31,55,16),(2,9,56,32),(3,25,49,10),(4,11,50,26),(5,27,51,12),(6,13,52,28),(7,29,53,14),(8,15,54,30),(17,59,33,42),(18,43,34,60),(19,61,35,44),(20,45,36,62),(21,63,37,46),(22,47,38,64),(23,57,39,48),(24,41,40,58)], [(1,59,55,42),(2,34,56,18),(3,61,49,44),(4,36,50,20),(5,63,51,46),(6,38,52,22),(7,57,53,48),(8,40,54,24),(9,43,32,60),(10,35,25,19),(11,45,26,62),(12,37,27,21),(13,47,28,64),(14,39,29,23),(15,41,30,58),(16,33,31,17)], [(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)]])
Q8⋊C8 is a maximal subgroup of
C42.455D4 C42.397D4 C42.399D4 C42.46D4 C42.373D4 C42.47D4 C42.401D4 Q8⋊M4(2) C42.374D4 D4⋊4M4(2) Q8⋊5M4(2) C42.316D4 C42.305D4 C42.52D4 C42.54D4 C8×SD16 C8×Q16 SD16⋊C8 Q16⋊5C8 C8⋊12SD16 C8⋊15SD16 C8⋊9Q16 D4.M4(2) Q8.M4(2) Q8⋊2M4(2) C8⋊9SD16 C8⋊M4(2) C42.181C23 Q8⋊D8 D4⋊SD16 Q8⋊SD16 Q8⋊6SD16 Q8⋊3D8 C42.189C23 C42.191C23 Q8⋊2SD16 D4⋊Q16 Q8⋊Q16 Q8.Q16 D4.3Q16 C42.199C23 C42.201C23 Q8.D8 Q8⋊3SD16 D4.5SD16 D4⋊3Q16 Q8⋊3Q16 Q8⋊4Q16 D4⋊4Q16 C42.211C23 Q8⋊4SD16 C42.213C23 Q8.SD16 C2.U2(𝔽3)
C4p.M4(2): C8⋊6Q16 C8.M4(2) C4.8Dic12 Dic6⋊2C8 C12.26Q16 Dic10⋊3C8 Dic10⋊4C8 C20.26Q16 ...
Q8⋊C8 is a maximal quotient of
C4⋊C4⋊C8 (C2×Q8)⋊C8 C42.46Q8 Dic10⋊1C8 Dic5.12Q16
C4p.Q16: Q8⋊C16 C8.17Q16 C4.8Dic12 Dic6⋊2C8 C12.26Q16 Dic10⋊3C8 Dic10⋊4C8 C20.26Q16 ...
Matrix representation of Q8⋊C8 ►in GL3(𝔽17) generated by
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 16 | 0 |
| 16 | 0 | 0 |
| 0 | 16 | 7 |
| 0 | 7 | 1 |
| 15 | 0 | 0 |
| 0 | 4 | 11 |
| 0 | 11 | 13 |
G:=sub<GL(3,GF(17))| [1,0,0,0,0,16,0,1,0],[16,0,0,0,16,7,0,7,1],[15,0,0,0,4,11,0,11,13] >;
Q8⋊C8 in GAP, Magma, Sage, TeX
Q_8\rtimes C_8
% in TeX
G:=Group("Q8:C8"); // GroupNames label
G:=SmallGroup(64,7);
// by ID
G=gap.SmallGroup(64,7);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,103,362,188,86,117]);
// Polycyclic
G:=Group<a,b,c|a^4=c^8=1,b^2=a^2,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^-1*b>;
// generators/relations
Export
Subgroup lattice of Q8⋊C8 in TeX
Character table of Q8⋊C8 in TeX