p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C8⋊Q8, C42.33C22, C4.8(C2×Q8), (C2×C4).46D4, C4⋊Q8.12C2, C2.9(C4⋊Q8), C4.Q8.3C2, C2.D8.8C2, C8⋊C4.2C2, C4⋊C4.25C22, (C2×C8).22C22, C42.C2.5C2, C2.23(C8⋊C22), (C2×C4).126C23, C22.122(C2×D4), C2.23(C8.C22), SmallGroup(64,182)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊Q8
G = < a,b,c | a8=b4=1, c2=b2, bab-1=a5, cac-1=a3, cbc-1=b-1 >
Character table of C8⋊Q8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | symplectic lifted from Q8, Schur index 2 |
| ρ12 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ15 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ16 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, 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 22 28)(2 38 23 25)(3 35 24 30)(4 40 17 27)(5 37 18 32)(6 34 19 29)(7 39 20 26)(8 36 21 31)(9 42 64 54)(10 47 57 51)(11 44 58 56)(12 41 59 53)(13 46 60 50)(14 43 61 55)(15 48 62 52)(16 45 63 49)
(1 50 22 46)(2 53 23 41)(3 56 24 44)(4 51 17 47)(5 54 18 42)(6 49 19 45)(7 52 20 48)(8 55 21 43)(9 37 64 32)(10 40 57 27)(11 35 58 30)(12 38 59 25)(13 33 60 28)(14 36 61 31)(15 39 62 26)(16 34 63 29)
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), (1,33,22,28)(2,38,23,25)(3,35,24,30)(4,40,17,27)(5,37,18,32)(6,34,19,29)(7,39,20,26)(8,36,21,31)(9,42,64,54)(10,47,57,51)(11,44,58,56)(12,41,59,53)(13,46,60,50)(14,43,61,55)(15,48,62,52)(16,45,63,49), (1,50,22,46)(2,53,23,41)(3,56,24,44)(4,51,17,47)(5,54,18,42)(6,49,19,45)(7,52,20,48)(8,55,21,43)(9,37,64,32)(10,40,57,27)(11,35,58,30)(12,38,59,25)(13,33,60,28)(14,36,61,31)(15,39,62,26)(16,34,63,29)>;
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), (1,33,22,28)(2,38,23,25)(3,35,24,30)(4,40,17,27)(5,37,18,32)(6,34,19,29)(7,39,20,26)(8,36,21,31)(9,42,64,54)(10,47,57,51)(11,44,58,56)(12,41,59,53)(13,46,60,50)(14,43,61,55)(15,48,62,52)(16,45,63,49), (1,50,22,46)(2,53,23,41)(3,56,24,44)(4,51,17,47)(5,54,18,42)(6,49,19,45)(7,52,20,48)(8,55,21,43)(9,37,64,32)(10,40,57,27)(11,35,58,30)(12,38,59,25)(13,33,60,28)(14,36,61,31)(15,39,62,26)(16,34,63,29) );
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)], [(1,33,22,28),(2,38,23,25),(3,35,24,30),(4,40,17,27),(5,37,18,32),(6,34,19,29),(7,39,20,26),(8,36,21,31),(9,42,64,54),(10,47,57,51),(11,44,58,56),(12,41,59,53),(13,46,60,50),(14,43,61,55),(15,48,62,52),(16,45,63,49)], [(1,50,22,46),(2,53,23,41),(3,56,24,44),(4,51,17,47),(5,54,18,42),(6,49,19,45),(7,52,20,48),(8,55,21,43),(9,37,64,32),(10,40,57,27),(11,35,58,30),(12,38,59,25),(13,33,60,28),(14,36,61,31),(15,39,62,26),(16,34,63,29)]])
C8⋊Q8 is a maximal subgroup of
C42.6C23 C42.7C23 C42.10C23 M4(2)⋊3Q8 M4(2)⋊4Q8 C42.390C23 C42.391C23 C42.423C23 C42.424C23 C42.425C23 C42.426C23 C42.57C23 C42.58C23 C42.59C23 C42.60C23 C42.61C23 C42.62C23 C42.63C23 C42.64C23 D8⋊4Q8 SD16⋊Q8 SD16⋊2Q8 SD16⋊3Q8 D8⋊5Q8
C42.D2p: C42.252D4 M4(2)⋊5Q8 M4(2)⋊6Q8 C42.255D4 C42.256D4 C42.257D4 C42.258D4 C42.286D4 ...
C4p⋊Q8.C2: C42.9C23 Q16⋊4Q8 Q16⋊5Q8 C24⋊3Q8 C24⋊4Q8 C40⋊3Q8 C40⋊4Q8 C56⋊3Q8 ...
C8⋊Q8 is a maximal quotient of
C42.26Q8 C4.(C4×Q8) C8⋊(C4⋊C4)
C42.D2p: C42.124D4 C42.125D4 C8⋊Dic6 C42.68D6 C42.76D6 C8⋊Dic10 C42.68D10 C42.76D10 ...
C4⋊C4.D2p: C4⋊C4⋊Q8 (C2×C8)⋊Q8 C2.(C8⋊Q8) (C2×C8).1Q8 C2.(C8⋊3Q8) (C2×C8).24Q8 C24⋊3Q8 C24⋊4Q8 ...
Matrix representation of C8⋊Q8 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 8 | 2 | 13 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 15 | 4 | 15 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 4 | 0 | 4 |
| 0 | 0 | 0 | 2 | 13 | 2 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 14 | 16 | 13 |
| 0 | 0 | 9 | 5 | 13 | 16 |
| 0 | 0 | 11 | 9 | 15 | 15 |
| 0 | 0 | 16 | 14 | 13 | 1 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,2,1,15,0,0,1,13,0,4,0,0,0,0,0,15],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,16,0,0,0,4,2,0,0,0,0,0,13,0,0,0,1,4,2,0],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,9,11,16,0,0,14,5,9,14,0,0,16,13,15,13,0,0,13,16,15,1] >;
C8⋊Q8 in GAP, Magma, Sage, TeX
C_8\rtimes Q_8
% in TeX
G:=Group("C8:Q8"); // GroupNames label
G:=SmallGroup(64,182);
// by ID
G=gap.SmallGroup(64,182);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,55,362,332,86,1444,88]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=1,c^2=b^2,b*a*b^-1=a^5,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C8⋊Q8 in TeX
Character table of C8⋊Q8 in TeX