p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊2Q8, C4.5D8, C4.4Q16, C42.85C22, (C4×C8).8C2, C4.7(C2×Q8), C2.11(C2×D8), (C2×C4).80D4, C4⋊Q8.11C2, C2.8(C4⋊Q8), C2.D8.7C2, C2.11(C2×Q16), C4⋊C4.24C22, (C2×C8).81C22, (C2×C4).125C23, C22.121(C2×D4), SmallGroup(64,181)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊2Q8
G = < a,b,c | a8=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >
Character table of C8⋊2Q8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | 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 | 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 | 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 | -1 | -1 | linear of order 2 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | √2 | -√2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ11 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | √2 | -√2 | √2 | -√2 | -√2 | orthogonal lifted from D8 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | -√2 | √2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | -√2 | √2 | -√2 | √2 | √2 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | symplectic lifted from Q8, Schur index 2 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | -2 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | √2 | -√2 | -√2 | √2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ20 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | -√2 | √2 | -√2 | √2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ21 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | √2 | -√2 | √2 | -√2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ22 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | -√2 | √2 | √2 | -√2 | √2 | symplectic lifted from Q16, Schur index 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 33 19 32)(2 34 20 25)(3 35 21 26)(4 36 22 27)(5 37 23 28)(6 38 24 29)(7 39 17 30)(8 40 18 31)(9 47 63 52)(10 48 64 53)(11 41 57 54)(12 42 58 55)(13 43 59 56)(14 44 60 49)(15 45 61 50)(16 46 62 51)
(1 51 19 46)(2 50 20 45)(3 49 21 44)(4 56 22 43)(5 55 23 42)(6 54 24 41)(7 53 17 48)(8 52 18 47)(9 40 63 31)(10 39 64 30)(11 38 57 29)(12 37 58 28)(13 36 59 27)(14 35 60 26)(15 34 61 25)(16 33 62 32)
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), (1,33,19,32)(2,34,20,25)(3,35,21,26)(4,36,22,27)(5,37,23,28)(6,38,24,29)(7,39,17,30)(8,40,18,31)(9,47,63,52)(10,48,64,53)(11,41,57,54)(12,42,58,55)(13,43,59,56)(14,44,60,49)(15,45,61,50)(16,46,62,51), (1,51,19,46)(2,50,20,45)(3,49,21,44)(4,56,22,43)(5,55,23,42)(6,54,24,41)(7,53,17,48)(8,52,18,47)(9,40,63,31)(10,39,64,30)(11,38,57,29)(12,37,58,28)(13,36,59,27)(14,35,60,26)(15,34,61,25)(16,33,62,32)>;
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), (1,33,19,32)(2,34,20,25)(3,35,21,26)(4,36,22,27)(5,37,23,28)(6,38,24,29)(7,39,17,30)(8,40,18,31)(9,47,63,52)(10,48,64,53)(11,41,57,54)(12,42,58,55)(13,43,59,56)(14,44,60,49)(15,45,61,50)(16,46,62,51), (1,51,19,46)(2,50,20,45)(3,49,21,44)(4,56,22,43)(5,55,23,42)(6,54,24,41)(7,53,17,48)(8,52,18,47)(9,40,63,31)(10,39,64,30)(11,38,57,29)(12,37,58,28)(13,36,59,27)(14,35,60,26)(15,34,61,25)(16,33,62,32) );
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)], [(1,33,19,32),(2,34,20,25),(3,35,21,26),(4,36,22,27),(5,37,23,28),(6,38,24,29),(7,39,17,30),(8,40,18,31),(9,47,63,52),(10,48,64,53),(11,41,57,54),(12,42,58,55),(13,43,59,56),(14,44,60,49),(15,45,61,50),(16,46,62,51)], [(1,51,19,46),(2,50,20,45),(3,49,21,44),(4,56,22,43),(5,55,23,42),(6,54,24,41),(7,53,17,48),(8,52,18,47),(9,40,63,31),(10,39,64,30),(11,38,57,29),(12,37,58,28),(13,36,59,27),(14,35,60,26),(15,34,61,25),(16,33,62,32)]])
C8⋊2Q8 is a maximal subgroup of
C4.10D16 C4.6Q32 D4.7D8 Q8⋊4Q16 D4⋊4Q16 C42.211C23 D4.1Q16 Q8.1Q16 C8.8SD16 C8.3Q16 C8.7Q16 C42.665C23 D8⋊1Q8 Q16⋊Q8 D8⋊Q8 C4.Q32 C4.4D16 C4.SD32 C8.12SD16 C42.364D4 M4(2)⋊4Q8 M4(2)⋊5Q8 C42.366D4 C42.367D4 C42.387C23 C42.259D4 C42.278D4 C42.280D4 C42.282D4 C42.283D4 C42.425C23 D4⋊5D8 D4⋊6Q16 C42.490C23 C42.491C23 C42.494C23 C42.498C23 Q8×D8 D8⋊6Q8 Q8×Q16 Q16⋊6Q8 SD16⋊Q8 SD16⋊2Q8
C4p.D8: C8⋊4Q16 C8.2D8 C24⋊8Q8 C12.17D8 C40⋊8Q8 C20.17D8 C56⋊8Q8 C28.17D8 ...
C8p⋊Q8: C16⋊2Q8 C16⋊3Q8 C16⋊Q8 C24⋊2Q8 C40⋊2Q8 C56⋊2Q8 ...
C2.(C8⋊pD4): C8⋊10SD16 C8⋊7Q16 C8⋊4SD16 C8⋊2Q16 C8⋊5D8 C8⋊5Q16 C8⋊5SD16 C42.666C23 ...
C8⋊2Q8 is a maximal quotient of
C42.59Q8 C8⋊5(C4⋊C4) C42.436D4 C4⋊C4⋊Q8 (C2×C8).1Q8
C8p⋊Q8: C16⋊2Q8 C16⋊3Q8 C16⋊Q8 C24⋊8Q8 C24⋊2Q8 C40⋊8Q8 C40⋊2Q8 C56⋊8Q8 ...
C4p.Q16: C16.5Q8 C12.17D8 C20.17D8 C28.17D8 ...
Matrix representation of C8⋊2Q8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 3 |
| 0 | 0 | 14 | 14 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 7 | 1 | 0 | 0 |
| 1 | 10 | 0 | 0 |
| 0 | 0 | 10 | 16 |
| 0 | 0 | 16 | 7 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,14,14,0,0,3,14],[0,16,0,0,1,0,0,0,0,0,0,1,0,0,16,0],[7,1,0,0,1,10,0,0,0,0,10,16,0,0,16,7] >;
C8⋊2Q8 in GAP, Magma, Sage, TeX
C_8\rtimes_2Q_8
% in TeX
G:=Group("C8:2Q8"); // GroupNames label
G:=SmallGroup(64,181);
// by ID
G=gap.SmallGroup(64,181);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,55,362,230,1444,88]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C8⋊2Q8 in TeX
Character table of C8⋊2Q8 in TeX