direct product, p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C22×Q8, C2.2C24, C4.7C23, C22.9C23, C23.14C22, (C22×C4).7C2, (C2×C4).30C22, SmallGroup(32,47)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×Q8
G = < a,b,c,d | a2=b2=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 78, all normal (4 characteristic)
C1, C2, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C22×Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, C22×Q8
Character table of C22×Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | |
| 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 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ11 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ12 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ13 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ14 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| 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 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ20 | 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 |
►Regular action on 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 21)(6 22)(7 23)(8 24)(13 19)(14 20)(15 17)(16 18)(25 30)(26 31)(27 32)(28 29)
(1 22)(2 23)(3 24)(4 21)(5 12)(6 9)(7 10)(8 11)(13 28)(14 25)(15 26)(16 27)(17 31)(18 32)(19 29)(20 30)
(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 3 19)(2 20 4 18)(5 27 7 25)(6 26 8 28)(9 15 11 13)(10 14 12 16)(21 32 23 30)(22 31 24 29)
G:=sub<Sym(32)| (1,9)(2,10)(3,11)(4,12)(5,21)(6,22)(7,23)(8,24)(13,19)(14,20)(15,17)(16,18)(25,30)(26,31)(27,32)(28,29), (1,22)(2,23)(3,24)(4,21)(5,12)(6,9)(7,10)(8,11)(13,28)(14,25)(15,26)(16,27)(17,31)(18,32)(19,29)(20,30), (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), (1,17,3,19)(2,20,4,18)(5,27,7,25)(6,26,8,28)(9,15,11,13)(10,14,12,16)(21,32,23,30)(22,31,24,29)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,21)(6,22)(7,23)(8,24)(13,19)(14,20)(15,17)(16,18)(25,30)(26,31)(27,32)(28,29), (1,22)(2,23)(3,24)(4,21)(5,12)(6,9)(7,10)(8,11)(13,28)(14,25)(15,26)(16,27)(17,31)(18,32)(19,29)(20,30), (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), (1,17,3,19)(2,20,4,18)(5,27,7,25)(6,26,8,28)(9,15,11,13)(10,14,12,16)(21,32,23,30)(22,31,24,29) );
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,21),(6,22),(7,23),(8,24),(13,19),(14,20),(15,17),(16,18),(25,30),(26,31),(27,32),(28,29)], [(1,22),(2,23),(3,24),(4,21),(5,12),(6,9),(7,10),(8,11),(13,28),(14,25),(15,26),(16,27),(17,31),(18,32),(19,29),(20,30)], [(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)], [(1,17,3,19),(2,20,4,18),(5,27,7,25),(6,26,8,28),(9,15,11,13),(10,14,12,16),(21,32,23,30),(22,31,24,29)]])
C22×Q8 is a maximal subgroup of
C23.67C23 C23⋊Q8 C23.78C23 C23.38D4 Q8⋊D4 C22⋊Q16 C23.32C23 C23.38C23 Q8⋊5D4 Q8⋊A4
C22×Q8 is a maximal quotient of
C23.37C23 C23⋊2Q8 C23.41C23 D4⋊3Q8 Q8⋊3Q8
Matrix representation of C22×Q8 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 3 |
| 0 | 0 | 1 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 2 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,1,1,0,0,3,4],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,4,2] >;
C22×Q8 in GAP, Magma, Sage, TeX
C_2^2\times Q_8
% in TeX
G:=Group("C2^2xQ8"); // GroupNames label
G:=SmallGroup(32,47);
// by ID
G=gap.SmallGroup(32,47);
# by ID
G:=PCGroup([5,-2,2,2,2,-2,80,181,86]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export