p-group, metabelian, nilpotent (class 3), monomial
Aliases: C8.5Q8, C42.84C22, (C4×C8).9C2, C4.6(C2×Q8), (C2×C4).59D4, C2.7(C4⋊Q8), C2.D8.6C2, C4.Q8.7C2, C2.19(C4○D8), C4⋊C4.23C22, (C2×C8).94C22, C42.C2.4C2, (C2×C4).124C23, C22.120(C2×D4), SmallGroup(64,180)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C8.5Q8
G = < a,b,c | a8=b4=1, c2=a4b2, ab=ba, cac-1=a3, cbc-1=a4b-1 >
Character table of C8.5Q8
| 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 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | -√2 | -√-2 | -√-2 | √2 | -√2 | √-2 | √2 | √-2 | complex lifted from C4○D8 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | -√2 | -√-2 | √-2 | -√2 | √2 | -√-2 | √2 | √-2 | complex lifted from C4○D8 |
| ρ17 | 2 | -2 | -2 | 2 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | √2 | -√-2 | -√-2 | -√2 | √2 | √-2 | -√2 | √-2 | complex lifted from C4○D8 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | √2 | -√-2 | √-2 | √2 | -√2 | -√-2 | -√2 | √-2 | complex lifted from C4○D8 |
| ρ19 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | -√2 | √-2 | -√-2 | -√2 | √2 | √-2 | √2 | -√-2 | complex lifted from C4○D8 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | √2 | √-2 | -√-2 | √2 | -√2 | √-2 | -√2 | -√-2 | complex lifted from C4○D8 |
| ρ21 | 2 | -2 | -2 | 2 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | √2 | √-2 | √-2 | -√2 | √2 | -√-2 | -√2 | -√-2 | complex lifted from C4○D8 |
| ρ22 | 2 | -2 | -2 | 2 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | -√2 | √-2 | √-2 | √2 | -√2 | -√-2 | √2 | -√-2 | complex lifted from C4○D8 |
►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 26)(2 34 20 27)(3 35 21 28)(4 36 22 29)(5 37 23 30)(6 38 24 31)(7 39 17 32)(8 40 18 25)(9 46 58 55)(10 47 59 56)(11 48 60 49)(12 41 61 50)(13 42 62 51)(14 43 63 52)(15 44 64 53)(16 45 57 54)
(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), (1,33,19,26)(2,34,20,27)(3,35,21,28)(4,36,22,29)(5,37,23,30)(6,38,24,31)(7,39,17,32)(8,40,18,25)(9,46,58,55)(10,47,59,56)(11,48,60,49)(12,41,61,50)(13,42,62,51)(14,43,63,52)(15,44,64,53)(16,45,57,54), (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), (1,33,19,26)(2,34,20,27)(3,35,21,28)(4,36,22,29)(5,37,23,30)(6,38,24,31)(7,39,17,32)(8,40,18,25)(9,46,58,55)(10,47,59,56)(11,48,60,49)(12,41,61,50)(13,42,62,51)(14,43,63,52)(15,44,64,53)(16,45,57,54), (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)], [(1,33,19,26),(2,34,20,27),(3,35,21,28),(4,36,22,29),(5,37,23,30),(6,38,24,31),(7,39,17,32),(8,40,18,25),(9,46,58,55),(10,47,59,56),(11,48,60,49),(12,41,61,50),(13,42,62,51),(14,43,63,52),(15,44,64,53),(16,45,57,54)], [(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)]])
C8.5Q8 is a maximal subgroup of
C8.1Q16 M4(2)⋊3Q8 M4(2)⋊4Q8 C42.386C23 C42.388C23 C42.389C23 C42.423C23 C42.424C23 C42.485C23 C42.486C23 C42.488C23 C42.492C23 C42.493C23 C42.496C23 D8⋊6Q8 SD16⋊4Q8 Q16⋊6Q8 SD16⋊3Q8 D8⋊5Q8 Q16⋊5Q8
C8p.Q8: C16.5Q8 C24.13Q8 C8.8Dic6 C8.6Dic6 C40.13Q8 C8.8Dic10 C8.6Dic10 C56.13Q8 ...
C42.D2p: C8.16Q16 D8.Q8 Q16.Q8 D8.2Q8 C8.22SD16 C8.13SD16 C8.14SD16 C16⋊Q8 ...
C8.5Q8 is a maximal quotient of
C42.60Q8 C8⋊7(C4⋊C4) C8⋊5(C4⋊C4)
C42.D2p: C42.437D4 C24.13Q8 C42.215D6 C40.13Q8 C42.215D10 C56.13Q8 C42.215D14 ...
C4⋊C4.D2p: C2.(C8⋊Q8) (C2×C8).24Q8 C8.8Dic6 C8.6Dic6 C8.8Dic10 C8.6Dic10 C56.8Q8 C56.4Q8 ...
Matrix representation of C8.5Q8 ►in GL4(𝔽17) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 13 | 0 | 0 |
| 9 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 13 | 16 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,15,0,0,0,0,9],[1,9,0,0,13,16,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,16,4,0,0,0,0,0,1,0,0,1,0] >;
C8.5Q8 in GAP, Magma, Sage, TeX
C_8._5Q_8
% in TeX
G:=Group("C8.5Q8"); // GroupNames label
G:=SmallGroup(64,180);
// by ID
G=gap.SmallGroup(64,180);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,247,362,86,1444,88]);
// Polycyclic
G:=Group<a,b,c|a^8=b^4=1,c^2=a^4*b^2,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=a^4*b^-1>;
// generators/relations
Export
Subgroup lattice of C8.5Q8 in TeX
Character table of C8.5Q8 in TeX