p-group, metabelian, nilpotent (class 2), monomial
Aliases: C8⋊4Q8, C4.3M4(2), C42.13C22, C4⋊C4.9C4, C4⋊C8.10C2, C2.6(C4×Q8), (C4×C8).15C2, (C2×Q8).7C4, (C4×Q8).4C2, C4.24(C2×Q8), C8⋊C4.5C2, C2.9(C8○D4), C4.55(C4○D4), (C2×C8).103C22, (C2×C4).157C23, C2.11(C2×M4(2)), C22.49(C22×C4), (C2×C4).30(C2×C4), SmallGroup(64,127)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8⋊4Q8
G = < a,b,c | a8=b4=1, c2=b2, ab=ba, cac-1=a5, cbc-1=b-1 >
Character table of C8⋊4Q8
| 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 | -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 | -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 | 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 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ9 | 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 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 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) |
| ρ20 | 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) |
| ρ21 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | -2i | 0 | 2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 2i | 0 | -2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| 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 | 0 | 2ζ85 | 0 | 2ζ83 | 0 | 2ζ87 | 0 | 2ζ8 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ26 | 2 | 2 | -2 | -2 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ8 | 0 | 2ζ87 | 0 | 2ζ83 | 0 | 2ζ85 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ27 | 2 | 2 | -2 | -2 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ87 | 0 | 2ζ8 | 0 | 2ζ85 | 0 | 2ζ83 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
| ρ28 | 2 | 2 | -2 | -2 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ83 | 0 | 2ζ85 | 0 | 2ζ8 | 0 | 2ζ87 | 0 | 0 | 0 | 0 | complex lifted from C8○D4 |
►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 28 21 38)(2 29 22 39)(3 30 23 40)(4 31 24 33)(5 32 17 34)(6 25 18 35)(7 26 19 36)(8 27 20 37)(9 42 58 56)(10 43 59 49)(11 44 60 50)(12 45 61 51)(13 46 62 52)(14 47 63 53)(15 48 64 54)(16 41 57 55)
(1 47 21 53)(2 44 22 50)(3 41 23 55)(4 46 24 52)(5 43 17 49)(6 48 18 54)(7 45 19 51)(8 42 20 56)(9 37 58 27)(10 34 59 32)(11 39 60 29)(12 36 61 26)(13 33 62 31)(14 38 63 28)(15 35 64 25)(16 40 57 30)
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), (1,28,21,38)(2,29,22,39)(3,30,23,40)(4,31,24,33)(5,32,17,34)(6,25,18,35)(7,26,19,36)(8,27,20,37)(9,42,58,56)(10,43,59,49)(11,44,60,50)(12,45,61,51)(13,46,62,52)(14,47,63,53)(15,48,64,54)(16,41,57,55), (1,47,21,53)(2,44,22,50)(3,41,23,55)(4,46,24,52)(5,43,17,49)(6,48,18,54)(7,45,19,51)(8,42,20,56)(9,37,58,27)(10,34,59,32)(11,39,60,29)(12,36,61,26)(13,33,62,31)(14,38,63,28)(15,35,64,25)(16,40,57,30)>;
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), (1,28,21,38)(2,29,22,39)(3,30,23,40)(4,31,24,33)(5,32,17,34)(6,25,18,35)(7,26,19,36)(8,27,20,37)(9,42,58,56)(10,43,59,49)(11,44,60,50)(12,45,61,51)(13,46,62,52)(14,47,63,53)(15,48,64,54)(16,41,57,55), (1,47,21,53)(2,44,22,50)(3,41,23,55)(4,46,24,52)(5,43,17,49)(6,48,18,54)(7,45,19,51)(8,42,20,56)(9,37,58,27)(10,34,59,32)(11,39,60,29)(12,36,61,26)(13,33,62,31)(14,38,63,28)(15,35,64,25)(16,40,57,30) );
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)], [(1,28,21,38),(2,29,22,39),(3,30,23,40),(4,31,24,33),(5,32,17,34),(6,25,18,35),(7,26,19,36),(8,27,20,37),(9,42,58,56),(10,43,59,49),(11,44,60,50),(12,45,61,51),(13,46,62,52),(14,47,63,53),(15,48,64,54),(16,41,57,55)], [(1,47,21,53),(2,44,22,50),(3,41,23,55),(4,46,24,52),(5,43,17,49),(6,48,18,54),(7,45,19,51),(8,42,20,56),(9,37,58,27),(10,34,59,32),(11,39,60,29),(12,36,61,26),(13,33,62,31),(14,38,63,28),(15,35,64,25),(16,40,57,30)]])
C8⋊4Q8 is a maximal subgroup of
C8.17Q16 C8⋊12SD16 C8⋊9Q16 D4.M4(2) Q8.M4(2) C8⋊9SD16 C8⋊M4(2) C8⋊SD16 C8⋊2SD16 C8.SD16 C8⋊Q16 C8⋊2Q16 C8.3Q16 C42.249C23 C42.251C23 C42.253C23 C42.255C23 C42.290C23 C42.291C23 C42.292C23 C42.293C23 C42.294C23 D4⋊6M4(2) C42.302C23 Q8.4M4(2) C42.696C23 C42.304C23 C42.305C23 C42.698C23 D4⋊8M4(2) C42.307C23 C42.308C23 C42.309C23 C42.310C23 C42.507C23 C42.508C23 C42.509C23 C42.510C23 C42.511C23 C42.512C23 C42.513C23 C42.514C23 C42.515C23 C42.516C23 C42.517C23 C42.518C23 D8⋊4Q8 SD16⋊Q8 SD16⋊2Q8 Q16⋊4Q8 SD16⋊3Q8 D8⋊5Q8 Q16⋊5Q8 C42.72C23 C42.73C23 C42.74C23 C42.75C23 C42.531C23 C42.532C23 C42.533C23
C4p.M4(2): C8⋊6Q16 C8.M4(2) C24⋊12Q8 C42.198D6 C42.210D6 C40⋊11Q8 C42.198D10 C42.210D10 ...
C2p.(C4×Q8): C42.286C23 C42.287C23 M4(2)⋊9Q8 Q8×M4(2) C24⋊Q8 C42.27D6 C40⋊Q8 Dic5.5M4(2) ...
C8⋊4Q8 is a maximal quotient of
C4⋊C8⋊13C4 C4⋊C8⋊14C4 C4⋊C4⋊3C8 (C2×C8).Q8 Dic5.M4(2) C20.M4(2) C20.6M4(2)
C42.D2p: C42.61Q8 C42.27Q8 C42.327D4 C42.120D4 C24⋊12Q8 C24⋊Q8 C42.27D6 C42.198D6 ...
Matrix representation of C8⋊4Q8 ►in GL4(𝔽17) generated by
| 0 | 1 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 13 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 15 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 8 |
| 0 | 0 | 15 | 7 |
G:=sub<GL(4,GF(17))| [0,13,0,0,1,0,0,0,0,0,0,13,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,15,0,0,9,0],[16,0,0,0,0,1,0,0,0,0,10,15,0,0,8,7] >;
C8⋊4Q8 in GAP, Magma, Sage, TeX
C_8\rtimes_4Q_8
% in TeX
G:=Group("C8:4Q8"); // GroupNames label
G:=SmallGroup(64,127);
// by ID
G=gap.SmallGroup(64,127);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,650,122,88]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C8⋊4Q8 in TeX
Character table of C8⋊4Q8 in TeX