direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4×Q8, C42.3C2, C22.8C23, C4○3(C4⋊C4), C4⋊C4.6C2, C4.4(C2×C4), C2.2(C2×Q8), (C2×Q8).5C2, C2.3(C4○D4), C2.5(C22×C4), (C2×C4).12C22, SmallGroup(32,26)
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, ac=ca, cbc-1=b-1 >
Character table of C4×Q8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 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 | trivial |
| ρ2 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 1 | -1 | 1 | -1 | -i | i | -i | i | i | 1 | i | i | -1 | -1 | -1 | -i | -i | 1 | -i | 1 | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | -1 | -i | i | -1 | 1 | 1 | i | i | -1 | -i | 1 | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | i | -i | i | -i | i | -1 | i | -i | -1 | 1 | 1 | -i | -i | -1 | i | 1 | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | i | -i | i | -i | -i | 1 | -i | -i | -1 | -1 | -1 | i | i | 1 | i | 1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | i | -i | i | -i | -i | -1 | i | i | 1 | -1 | 1 | -i | i | 1 | -i | -1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | i | -i | i | -i | i | 1 | -i | i | 1 | 1 | -1 | i | -i | -1 | -i | -1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | -i | i | -i | i | -i | 1 | i | -i | 1 | 1 | -1 | -i | i | -1 | i | -1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | -i | i | -i | i | i | -1 | -i | -i | 1 | -1 | 1 | i | -i | 1 | i | -1 | linear of order 4 |
| ρ17 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | -2 | 2 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►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 17 9 14)(2 18 10 15)(3 19 11 16)(4 20 12 13)(5 27 30 24)(6 28 31 21)(7 25 32 22)(8 26 29 23)
(1 28 9 21)(2 25 10 22)(3 26 11 23)(4 27 12 24)(5 13 30 20)(6 14 31 17)(7 15 32 18)(8 16 29 19)
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), (1,17,9,14)(2,18,10,15)(3,19,11,16)(4,20,12,13)(5,27,30,24)(6,28,31,21)(7,25,32,22)(8,26,29,23), (1,28,9,21)(2,25,10,22)(3,26,11,23)(4,27,12,24)(5,13,30,20)(6,14,31,17)(7,15,32,18)(8,16,29,19)>;
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), (1,17,9,14)(2,18,10,15)(3,19,11,16)(4,20,12,13)(5,27,30,24)(6,28,31,21)(7,25,32,22)(8,26,29,23), (1,28,9,21)(2,25,10,22)(3,26,11,23)(4,27,12,24)(5,13,30,20)(6,14,31,17)(7,15,32,18)(8,16,29,19) );
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)], [(1,17,9,14),(2,18,10,15),(3,19,11,16),(4,20,12,13),(5,27,30,24),(6,28,31,21),(7,25,32,22),(8,26,29,23)], [(1,28,9,21),(2,25,10,22),(3,26,11,23),(4,27,12,24),(5,13,30,20),(6,14,31,17),(7,15,32,18),(8,16,29,19)]])
C4×Q8 is a maximal subgroup of
Q8⋊C8 SD16⋊C4 C8⋊4Q8 C4⋊SD16 C4⋊2Q16 Q8.D4 Q8⋊Q8 C4.Q16 Q8.Q8 C23.32C23 C23.33C23 C23.36C23 C23.37C23 C22.35C24 C22.36C24 Q8⋊5D4 Q8⋊6D4 C22.46C24 D4⋊3Q8 C22.50C24 Q8⋊3Q8 C22.53C24
C4p.(C2×C4): Q16⋊C4 Dic6⋊C4 Dic5⋊3Q8 Dic7⋊3Q8 Dic22⋊C4 Dic13⋊3Q8 ...
C4×Q8 is a maximal quotient of
C8⋊4Q8
(C2×C4).D2p: C23.63C23 C23.65C23 C23.67C23 Dic6⋊C4 Dic5⋊3Q8 Dic7⋊3Q8 Dic22⋊C4 Dic13⋊3Q8 ...
Matrix representation of C4×Q8 ►in GL3(𝔽5) generated by
| 2 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 2 |
| 4 | 0 | 0 |
| 0 | 0 | 3 |
| 0 | 3 | 0 |
G:=sub<GL(3,GF(5))| [2,0,0,0,3,0,0,0,3],[1,0,0,0,3,0,0,0,2],[4,0,0,0,0,3,0,3,0] >;
C4×Q8 in GAP, Magma, Sage, TeX
C_4\times Q_8
% in TeX
G:=Group("C4xQ8"); // GroupNames label
G:=SmallGroup(32,26);
// by ID
G=gap.SmallGroup(32,26);
# by ID
G:=PCGroup([5,-2,2,2,-2,2,80,101,46,102]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C4×Q8 in TeX
Character table of C4×Q8 in TeX