direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C4⋊C4, C22.3Q8, C22.13D4, C22.5C23, C23.13C22, C4⋊2(C2×C4), (C2×C4)⋊3C4, C2.2(C2×D4), C2.1(C2×Q8), C2.2(C22×C4), (C2×C4).9C22, (C22×C4).3C2, C22.10(C2×C4), SmallGroup(32,23)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4⋊C4
G = < a,b,c | a2=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C2×C4⋊C4
| 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 | -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 | 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 | 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
(1 7)(2 8)(3 5)(4 6)(9 24)(10 21)(11 22)(12 23)(13 17)(14 18)(15 19)(16 20)(25 30)(26 31)(27 32)(28 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 19 24 29)(2 18 21 32)(3 17 22 31)(4 20 23 30)(5 13 11 26)(6 16 12 25)(7 15 9 28)(8 14 10 27)
G:=sub<Sym(32)| (1,7)(2,8)(3,5)(4,6)(9,24)(10,21)(11,22)(12,23)(13,17)(14,18)(15,19)(16,20)(25,30)(26,31)(27,32)(28,29), (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), (1,19,24,29)(2,18,21,32)(3,17,22,31)(4,20,23,30)(5,13,11,26)(6,16,12,25)(7,15,9,28)(8,14,10,27)>;
G:=Group( (1,7)(2,8)(3,5)(4,6)(9,24)(10,21)(11,22)(12,23)(13,17)(14,18)(15,19)(16,20)(25,30)(26,31)(27,32)(28,29), (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), (1,19,24,29)(2,18,21,32)(3,17,22,31)(4,20,23,30)(5,13,11,26)(6,16,12,25)(7,15,9,28)(8,14,10,27) );
G=PermutationGroup([[(1,7),(2,8),(3,5),(4,6),(9,24),(10,21),(11,22),(12,23),(13,17),(14,18),(15,19),(16,20),(25,30),(26,31),(27,32),(28,29)], [(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)], [(1,19,24,29),(2,18,21,32),(3,17,22,31),(4,20,23,30),(5,13,11,26),(6,16,12,25),(7,15,9,28),(8,14,10,27)]])
C2×C4⋊C4 is a maximal subgroup of
C22.M4(2) C22.4Q16 C22.C42 C23.7Q8 C42⋊8C4 C42⋊9C4 C23.8Q8 C23.63C23 C24.C22 C23.65C23 C24.3C22 C23.67C23 C23.10D4 C23.78C23 C23.Q8 C23.11D4 C23.81C23 C23.4Q8 C23.83C23 C23.36D4 M4(2)⋊C4 C22.D8 C23.46D4 C23.47D4 C23.48D4 C2×C4×D4 C2×C4×Q8 C23.33C23 C22.31C24 C22.33C24 C23.41C23 D4⋊6D4 C22.46C24 C22.47C24 D4⋊3Q8
C2×C4⋊C4 is a maximal quotient of
C23.7Q8 C42⋊8C4 C42⋊9C4 C23.8Q8 C23.65C23 C4⋊M4(2) C42.6C22 C23.25D4 M4(2)⋊C4 M4(2).C4
Matrix representation of C2×C4⋊C4 ►in GL4(𝔽5) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 1 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0],[3,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0] >;
C2×C4⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes C_4
% in TeX
G:=Group("C2xC4:C4"); // GroupNames label
G:=SmallGroup(32,23);
// by ID
G=gap.SmallGroup(32,23);
# by ID
G:=PCGroup([5,-2,2,2,-2,2,80,101,46]);
// Polycyclic
G:=Group<a,b,c|a^2=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C2×C4⋊C4 in TeX
Character table of C2×C4⋊C4 in TeX