p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2.1C42, C22.7D4, C22.2Q8, C23.12C22, (C2×C4)⋊2C4, C2.1(C4⋊C4), C22.6(C2×C4), (C22×C4).1C2, C2.1(C22⋊C4), SmallGroup(32,2)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2.C42
G = < a,b,c | a2=b4=c4=1, cbc-1=ab=ba, ac=ca >
Character table of C2.C42
| 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 | -i | -i | i | -i | i | -1 | 1 | -1 | i | -i | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -i | -1 | -i | i | i | -i | i | i | -i | 1 | 1 | -1 | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | i | -i | i | -i | i | 1 | -1 | 1 | -i | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | i | 1 | -i | i | i | -i | -i | -i | i | -1 | -1 | 1 | linear of order 4 |
| ρ9 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -i | 1 | i | -i | -i | i | i | i | -i | -1 | -1 | 1 | linear of order 4 |
| ρ10 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | i | i | 1 | 1 | -1 | -1 | i | -i | -i | i | -i | -i | linear of order 4 |
| ρ11 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | i | -i | -1 | -1 | 1 | 1 | i | -i | -i | -i | i | i | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | i | i | -i | i | -i | -1 | 1 | -1 | -i | i | -i | linear of order 4 |
| ρ13 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -i | -i | 1 | 1 | -1 | -1 | -i | i | i | -i | i | i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | i | -1 | i | -i | -i | i | -i | -i | i | 1 | 1 | -1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -i | i | -i | i | -i | 1 | -1 | 1 | i | -i | i | linear of order 4 |
| ρ16 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -i | i | -1 | -1 | 1 | 1 | -i | i | i | i | -i | -i | 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 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| 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
(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 30)(18 31)(19 32)(20 29)
(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 31 27 11)(2 19 28 8)(3 29 25 9)(4 17 26 6)(5 21 20 13)(7 23 18 15)(10 22 30 14)(12 24 32 16)
G:=sub<Sym(32)| (1,23)(2,24)(3,21)(4,22)(5,9)(6,10)(7,11)(8,12)(13,25)(14,26)(15,27)(16,28)(17,30)(18,31)(19,32)(20,29), (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), (1,31,27,11)(2,19,28,8)(3,29,25,9)(4,17,26,6)(5,21,20,13)(7,23,18,15)(10,22,30,14)(12,24,32,16)>;
G:=Group( (1,23)(2,24)(3,21)(4,22)(5,9)(6,10)(7,11)(8,12)(13,25)(14,26)(15,27)(16,28)(17,30)(18,31)(19,32)(20,29), (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), (1,31,27,11)(2,19,28,8)(3,29,25,9)(4,17,26,6)(5,21,20,13)(7,23,18,15)(10,22,30,14)(12,24,32,16) );
G=PermutationGroup([[(1,23),(2,24),(3,21),(4,22),(5,9),(6,10),(7,11),(8,12),(13,25),(14,26),(15,27),(16,28),(17,30),(18,31),(19,32),(20,29)], [(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)], [(1,31,27,11),(2,19,28,8),(3,29,25,9),(4,17,26,6),(5,21,20,13),(7,23,18,15),(10,22,30,14),(12,24,32,16)]])
C2.C42 is a maximal subgroup of
C22.SD16 C23.31D4 C4×C22⋊C4 C4×C4⋊C4 C23.7Q8 C23.34D4 C42⋊8C4 C42⋊5C4 C23.8Q8 C23.23D4 C23.63C23 C24.C22 C23.65C23 C23.67C23 C23⋊2D4 C23⋊Q8 C23.10D4 C23.78C23 C23.Q8 C23.11D4 C23.81C23 C23.4Q8 C23.83C23 C23.84C23 C23.3A4 C62.D4 C62.Q8 (C6×C12)⋊2C4
C2p.C42: C42⋊4C4 C6.C42 C10.10C42 D10.3Q8 C14.C42 C22.C42 C26.10C42 D26.Q8 ...
C2.C42 is a maximal quotient of
C22.7C42 C23.9D4 C22.C42 M4(2)⋊4C4 C62.D4 C62.Q8 (C6×C12)⋊2C4
C2p.C42: C4.9C42 C4.10C42 C42⋊6C4 C22.4Q16 C4.C42 C6.C42 C10.10C42 D10.3Q8 ...
Matrix representation of C2.C42 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[3,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[4,0,0,0,0,2,0,0,0,0,0,1,0,0,1,0] >;
C2.C42 in GAP, Magma, Sage, TeX
C_2.C_4^2
% in TeX
G:=Group("C2.C4^2"); // GroupNames label
G:=SmallGroup(32,2);
// by ID
G=gap.SmallGroup(32,2);
# by ID
G:=PCGroup([5,-2,2,-2,2,2,40,61,86]);
// Polycyclic
G:=Group<a,b,c|a^2=b^4=c^4=1,c*b*c^-1=a*b=b*a,a*c=c*a>;
// generators/relations
Export
Subgroup lattice of C2.C42 in TeX
Character table of C2.C42 in TeX