p-group, metabelian, nilpotent (class 2), monomial
Aliases: C4.3Q8, C42.4C2, C22.15C23, C4⋊C4.4C2, C2.4(C2×Q8), C2.8(C4○D4), (C2×C4).3C22, SmallGroup(32,32)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.C2
G = < a,b,c | a4=b4=1, c2=b2, ab=ba, cac-1=ab2, cbc-1=a2b >
Character table of C42.C2
| 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 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ10 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2i | 0 | -2i | 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 15 6)(2 18 16 7)(3 19 13 8)(4 20 14 5)(9 29 24 28)(10 30 21 25)(11 31 22 26)(12 32 23 27)
(1 26 15 31)(2 32 16 27)(3 28 13 29)(4 30 14 25)(5 12 20 23)(6 24 17 9)(7 10 18 21)(8 22 19 11)
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), (1,17,15,6)(2,18,16,7)(3,19,13,8)(4,20,14,5)(9,29,24,28)(10,30,21,25)(11,31,22,26)(12,32,23,27), (1,26,15,31)(2,32,16,27)(3,28,13,29)(4,30,14,25)(5,12,20,23)(6,24,17,9)(7,10,18,21)(8,22,19,11)>;
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), (1,17,15,6)(2,18,16,7)(3,19,13,8)(4,20,14,5)(9,29,24,28)(10,30,21,25)(11,31,22,26)(12,32,23,27), (1,26,15,31)(2,32,16,27)(3,28,13,29)(4,30,14,25)(5,12,20,23)(6,24,17,9)(7,10,18,21)(8,22,19,11) );
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)], [(1,17,15,6),(2,18,16,7),(3,19,13,8),(4,20,14,5),(9,29,24,28),(10,30,21,25),(11,31,22,26),(12,32,23,27)], [(1,26,15,31),(2,32,16,27),(3,28,13,29),(4,30,14,25),(5,12,20,23),(6,24,17,9),(7,10,18,21),(8,22,19,11)]])
C42.C2 is a maximal subgroup of
C23.36C23 C23.37C23 C22.33C24 C22.34C24 C22.35C24 C23.41C23 C22.46C24 C22.47C24 D4⋊3Q8 Q8⋊3Q8 C22.56C24 C22.57C24 C22.58C24 C4.3PSU3(𝔽2)
C4p.Q8: C8.5Q8 C12.6Q8 C4.Dic6 C20.6Q8 C4.Dic10 C28.6Q8 C28.3Q8 C44.6Q8 ...
(C2×C4).D2p: C42.2C22 D4.Q8 Q8.Q8 C42.78C22 C42.29C22 C42.30C22 C8⋊Q8 Dic3.Q8 ...
C42.C2 is a maximal quotient of
C4.3PSU3(𝔽2)
(C2×C4).D2p: C42⋊8C4 C23.63C23 C23.65C23 C23.81C23 C23.83C23 C12.6Q8 Dic3.Q8 C4.Dic6 ...
Matrix representation of C42.C2 ►in GL4(𝔽5) generated by
| 1 | 3 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 2 | 0 | 0 |
| 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(5))| [1,0,0,0,3,4,0,0,0,0,2,0,0,0,0,3],[3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,2],[4,4,0,0,2,1,0,0,0,0,0,4,0,0,1,0] >;
C42.C2 in GAP, Magma, Sage, TeX
C_4^2.C_2
% in TeX
G:=Group("C4^2.C2"); // GroupNames label
G:=SmallGroup(32,32);
// by ID
G=gap.SmallGroup(32,32);
# by ID
G:=PCGroup([5,-2,2,2,-2,2,40,101,86,302,42]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,c*b*c^-1=a^2*b>;
// generators/relations
Export
Subgroup lattice of C42.C2 in TeX
Character table of C42.C2 in TeX