p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42:1C2, C22.6C23, C23.7C22, C4o(C4:C4), C4:C4:6C2, (C2xC4):4C4, C4.9(C2xC4), C4o(C22:C4), C2.1(C4oD4), C22:C4.3C2, C22.5(C2xC4), C2.3(C22xC4), (C22xC4).4C2, (C2xC4).10C22, SmallGroup(32,24)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42:C2
G = < a,b,c | a4=b4=c2=1, ab=ba, cac=ab2, bc=cb >
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C4oD4, C42:C2
Character table of C42:C2
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 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 | -i | i | i | -i | -1 | 1 | i | -i | i | -i | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | i | i | i | 1 | -1 | -i | -i | -i | -i | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | i | i | -i | -i | 1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -i | -i | i | i | 1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | -i | -i | -i | 1 | -1 | i | i | i | i | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | -i | -i | i | -1 | 1 | -i | i | -i | i | linear of order 4 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
►On 16 points - transitive group 16T17
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9 15 6)(2 10 16 7)(3 11 13 8)(4 12 14 5)
(1 3)(2 14)(4 16)(5 10)(6 8)(7 12)(9 11)(13 15)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,15,6)(2,10,16,7)(3,11,13,8)(4,12,14,5), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,9,15,6)(2,10,16,7)(3,11,13,8)(4,12,14,5), (1,3)(2,14)(4,16)(5,10)(6,8)(7,12)(9,11)(13,15) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,9,15,6),(2,10,16,7),(3,11,13,8),(4,12,14,5)], [(1,3),(2,14),(4,16),(5,10),(6,8),(7,12),(9,11),(13,15)]])
G:=TransitiveGroup(16,17);
C42:C2 is a maximal subgroup of
C4xC4oD4 C22.11C24 C23.32C23 C23.33C23 C22.19C24 C23.36C23 C23.37C23 C22.29C24 C22.34C24 C22.35C24 C22.36C24 C23.41C23 C22.45C24 C22.46C24 C22.47C24 C22.49C24 C22.50C24 D10.C23 (C6xC12):5C4 D26.C23
C23.D2p: C4.9C42 C42:6C4 C23.24D4 C23.37D4 C23.38D4 C42:C22 C23.25D4 M4(2):C4 ...
(C4xC4p):C2: C8o2M4(2) C42.7C22 C42:2S3 C42:D5 C42:D7 C42:D11 C42:D13 ...
(C2xC4).D2p: M4(2):4C4 C23.C23 C42.6C22 C23.38C23 C4:C4:7S3 C4:C4:7D5 C4:C4:7D7 C4:C4:7D11 ...
C42:C2 is a maximal quotient of
C42:4C4 C42:5C4 C42.6C4 D10.C23 (C6xC12):5C4 D26.C23
C23.D2p: C23.7Q8 C23.34D4 C23.16D6 C23.26D6 C23.11D10 C23.21D10 C23.11D14 C23.21D14 ...
(C4xC4p):C2: C42.12C4 C42.7C22 C42:2S3 C42:D5 C42:D7 C42:D11 C42:D13 ...
(C2xC4).D2p: C4xC22:C4 C4xC4:C4 C42:8C4 C23.63C23 C24.C22 C4:C4:7S3 C4:C4:7D5 C4:C4:7D7 ...
Matrix representation of C42:C2 ►in GL3(F5) generated by
| 2 | 0 | 0 |
| 0 | 2 | 3 |
| 0 | 0 | 3 |
| 4 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 4 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 2 | 4 |
G:=sub<GL(3,GF(5))| [2,0,0,0,2,0,0,3,3],[4,0,0,0,2,0,0,0,2],[4,0,0,0,1,2,0,0,4] >;
C42:C2 in GAP, Magma, Sage, TeX
C_4^2\rtimes C_2
% in TeX
G:=Group("C4^2:C2"); // GroupNames label
G:=SmallGroup(32,24);
// by ID
G=gap.SmallGroup(32,24);
# by ID
G:=PCGroup([5,-2,2,2,-2,2,80,101,42]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^2=1,a*b=b*a,c*a*c=a*b^2,b*c=c*b>;
// generators/relations
Export
Subgroup lattice of C42:C2 in TeX
Character table of C42:C2 in TeX