p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C4⋊Q8, C4.5D4, C42.5C2, C22.18C23, C4⋊C4.5C2, C2.5(C2×Q8), C2.10(C2×D4), (C2×Q8).3C2, (C2×C4).5C22, SmallGroup(32,35)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4⋊Q8
G = < a,b,c | a4=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >
Character table of C4⋊Q8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | |
| size | 1 | 1 | 1 | 1 | 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 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 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 | linear of order 2 |
| ρ6 | 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 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ12 | 2 | -2 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
►Regular action on 32 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)
(1 7 22 10)(2 8 23 11)(3 5 24 12)(4 6 21 9)(13 20 25 29)(14 17 26 30)(15 18 27 31)(16 19 28 32)
(1 14 22 26)(2 13 23 25)(3 16 24 28)(4 15 21 27)(5 32 12 19)(6 31 9 18)(7 30 10 17)(8 29 11 20)
G:=sub<Sym(32)| (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), (1,7,22,10)(2,8,23,11)(3,5,24,12)(4,6,21,9)(13,20,25,29)(14,17,26,30)(15,18,27,31)(16,19,28,32), (1,14,22,26)(2,13,23,25)(3,16,24,28)(4,15,21,27)(5,32,12,19)(6,31,9,18)(7,30,10,17)(8,29,11,20)>;
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), (1,7,22,10)(2,8,23,11)(3,5,24,12)(4,6,21,9)(13,20,25,29)(14,17,26,30)(15,18,27,31)(16,19,28,32), (1,14,22,26)(2,13,23,25)(3,16,24,28)(4,15,21,27)(5,32,12,19)(6,31,9,18)(7,30,10,17)(8,29,11,20) );
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)], [(1,7,22,10),(2,8,23,11),(3,5,24,12),(4,6,21,9),(13,20,25,29),(14,17,26,30),(15,18,27,31),(16,19,28,32)], [(1,14,22,26),(2,13,23,25),(3,16,24,28),(4,15,21,27),(5,32,12,19),(6,31,9,18),(7,30,10,17),(8,29,11,20)]])
C4⋊Q8 is a maximal subgroup of
C4.10D8 C4.6Q16 C42.3C4 D4.10D4 D4.D4 C4⋊2Q16 D4⋊Q8 Q8⋊Q8 D4⋊2Q8 C4.Q16 C4.4D8 C4.SD16 C42.28C22 C42.30C22 C8⋊5D4 C22.26C24 C23.37C23 C23.38C23 C22.35C24 C22.36C24 C23.41C23 D4⋊6D4 D4×Q8 D4⋊3Q8 C22.49C24 C22.50C24 Q8⋊3Q8 Q82 C22.57C24 C32⋊C4⋊Q8 C4⋊PSU3(𝔽2)
C4p⋊Q8: C8⋊3Q8 C8⋊2Q8 C8⋊Q8 C12⋊2Q8 C12⋊Q8 C20⋊2Q8 C20⋊Q8 C28⋊2Q8 ...
C4p.D4: C4⋊Q16 C8.2D4 Dic3⋊Q8 Dic5⋊Q8 Dic7⋊Q8 Dic11⋊Q8 Dic13⋊Q8 ...
C4⋊Q8 is a maximal quotient of
C32⋊C4⋊Q8 C4⋊PSU3(𝔽2)
C4p⋊Q8: C8⋊3Q8 C8⋊2Q8 C8⋊Q8 C12⋊2Q8 C12⋊Q8 C20⋊2Q8 C20⋊Q8 C28⋊2Q8 ...
(C2×C4).D2p: C42⋊9C4 C23.65C23 C23.67C23 C23.78C23 C23.81C23 C8.5Q8 Dic3⋊Q8 Dic5⋊Q8 ...
Matrix representation of C4⋊Q8 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0],[3,0,0,0,0,2,0,0,0,0,4,0,0,0,0,4],[0,4,0,0,1,0,0,0,0,0,4,0,0,0,0,1] >;
C4⋊Q8 in GAP, Magma, Sage, TeX
C_4\rtimes Q_8
% in TeX
G:=Group("C4:Q8"); // GroupNames label
G:=SmallGroup(32,35);
// by ID
G=gap.SmallGroup(32,35);
# by ID
G:=PCGroup([5,-2,2,2,-2,2,40,101,46,302,72]);
// Polycyclic
G:=Group<a,b,c|a^4=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 C4⋊Q8 in TeX
Character table of C4⋊Q8 in TeX