p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8⋊3Q8, C4.5SD16, C42.83C22, C4.5(C2×Q8), (C4×C8).12C2, (C2×C4).79D4, C4⋊Q8.10C2, C2.6(C4⋊Q8), C4.Q8.6C2, C4⋊C4.22C22, (C2×C8).93C22, C2.17(C2×SD16), (C2×C4).123C23, C22.119(C2×D4), SmallGroup(64,179)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊3Q8
G = < a,b,c | a8=b4=1, c2=b2, ab=ba, cac-1=a3, cbc-1=b-1 >
Character table of C8⋊3Q8
| 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 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | √-2 | -√-2 | -√-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | √-2 | √-2 | -√-2 | -√-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | -√-2 | √-2 | √-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | -√-2 | -√-2 | √-2 | √-2 | -√-2 | √-2 | complex lifted from SD16 |
| ρ19 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | √-2 | -√-2 | √-2 | -√-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | √-2 | -√-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ21 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | -√-2 | √-2 | -√-2 | √-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | -√-2 | √-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
►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 37 23 26)(2 38 24 27)(3 39 17 28)(4 40 18 29)(5 33 19 30)(6 34 20 31)(7 35 21 32)(8 36 22 25)(9 46 62 51)(10 47 63 52)(11 48 64 53)(12 41 57 54)(13 42 58 55)(14 43 59 56)(15 44 60 49)(16 45 61 50)
(1 51 23 46)(2 54 24 41)(3 49 17 44)(4 52 18 47)(5 55 19 42)(6 50 20 45)(7 53 21 48)(8 56 22 43)(9 37 62 26)(10 40 63 29)(11 35 64 32)(12 38 57 27)(13 33 58 30)(14 36 59 25)(15 39 60 28)(16 34 61 31)
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), (1,37,23,26)(2,38,24,27)(3,39,17,28)(4,40,18,29)(5,33,19,30)(6,34,20,31)(7,35,21,32)(8,36,22,25)(9,46,62,51)(10,47,63,52)(11,48,64,53)(12,41,57,54)(13,42,58,55)(14,43,59,56)(15,44,60,49)(16,45,61,50), (1,51,23,46)(2,54,24,41)(3,49,17,44)(4,52,18,47)(5,55,19,42)(6,50,20,45)(7,53,21,48)(8,56,22,43)(9,37,62,26)(10,40,63,29)(11,35,64,32)(12,38,57,27)(13,33,58,30)(14,36,59,25)(15,39,60,28)(16,34,61,31)>;
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), (1,37,23,26)(2,38,24,27)(3,39,17,28)(4,40,18,29)(5,33,19,30)(6,34,20,31)(7,35,21,32)(8,36,22,25)(9,46,62,51)(10,47,63,52)(11,48,64,53)(12,41,57,54)(13,42,58,55)(14,43,59,56)(15,44,60,49)(16,45,61,50), (1,51,23,46)(2,54,24,41)(3,49,17,44)(4,52,18,47)(5,55,19,42)(6,50,20,45)(7,53,21,48)(8,56,22,43)(9,37,62,26)(10,40,63,29)(11,35,64,32)(12,38,57,27)(13,33,58,30)(14,36,59,25)(15,39,60,28)(16,34,61,31) );
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)], [(1,37,23,26),(2,38,24,27),(3,39,17,28),(4,40,18,29),(5,33,19,30),(6,34,20,31),(7,35,21,32),(8,36,22,25),(9,46,62,51),(10,47,63,52),(11,48,64,53),(12,41,57,54),(13,42,58,55),(14,43,59,56),(15,44,60,49),(16,45,61,50)], [(1,51,23,46),(2,54,24,41),(3,49,17,44),(4,52,18,47),(5,55,19,42),(6,50,20,45),(7,53,21,48),(8,56,22,43),(9,37,62,26),(10,40,63,29),(11,35,64,32),(12,38,57,27),(13,33,58,30),(14,36,59,25),(15,39,60,28),(16,34,61,31)]])
C8⋊3Q8 is a maximal subgroup of
C8⋊8SD16 C8⋊5SD16 C8⋊6SD16 C42.665C23 C42.667C23 C8⋊3Q16 C42.364D4 M4(2)⋊5Q8 C42.365D4 C42.261D4 C42.262D4 C42.279D4 C42.280D4 C42.281D4 C42.283D4
C4p.SD16: C8⋊5Q16 C82⋊12C2 C8.9SD16 D8⋊3Q8 C24⋊9Q8 C12.SD16 C40⋊9Q8 C20.SD16 ...
C4⋊C4.D2p: Q8⋊4SD16 C42.213C23 Q8.SD16 D4⋊4SD16 C8⋊11SD16 C8⋊8Q16 C8⋊3SD16 C8⋊Q16 ...
C8⋊3Q8 is a maximal quotient of
C42.58Q8 C8⋊7(C4⋊C4)
C42.D2p: C42.436D4 C24⋊9Q8 C12.SD16 C40⋊9Q8 C20.SD16 C56⋊9Q8 C28.SD16 ...
C4⋊C4.D2p: (C2×C8)⋊Q8 C2.(C8⋊3Q8) C24⋊5Q8 C40⋊5Q8 C56⋊5Q8 ...
Matrix representation of C8⋊3Q8 ►in GL4(𝔽17) generated by
| 12 | 5 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 12 | 12 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 11 |
| 0 | 0 | 11 | 13 |
G:=sub<GL(4,GF(17))| [12,12,0,0,5,12,0,0,0,0,12,12,0,0,5,12],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,4,0,0,0,0,4,11,0,0,11,13] >;
C8⋊3Q8 in GAP, Magma, Sage, TeX
C_8\rtimes_3Q_8
% in TeX
G:=Group("C8:3Q8"); // GroupNames label
G:=SmallGroup(64,179);
// by ID
G=gap.SmallGroup(64,179);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,55,362,86,1444,88]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C8⋊3Q8 in TeX
Character table of C8⋊3Q8 in TeX