p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42⋊2C2, C23.4C22, C22.16C23, C4⋊C4⋊5C2, C2.9(C4○D4), C22⋊C4.2C2, (C2×C4).4C22, SmallGroup(32,33)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊2C2
G = < a,b,c | a4=b4=c2=1, ab=ba, cac=ab2, cbc=a2b-1 >
Character table of C42⋊2C2
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 16 points - transitive group 16T27
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 13 5 11)(2 14 6 12)(3 15 7 9)(4 16 8 10)
(2 6)(4 8)(9 13)(10 12)(11 15)(14 16)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,13,5,11)(2,14,6,12)(3,15,7,9)(4,16,8,10), (2,6)(4,8)(9,13)(10,12)(11,15)(14,16)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,13,5,11)(2,14,6,12)(3,15,7,9)(4,16,8,10), (2,6)(4,8)(9,13)(10,12)(11,15)(14,16) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,13,5,11),(2,14,6,12),(3,15,7,9),(4,16,8,10)], [(2,6),(4,8),(9,13),(10,12),(11,15),(14,16)]])
G:=TransitiveGroup(16,27);
C42⋊2C2 is a maximal subgroup of
C22.32C24 C22.33C24 C22.45C24 C22.54C24 C22.57C24 C42⋊C6
C2p.(C4○D4): C23.36C23 C22.35C24 C22.36C24 C22.46C24 C22.47C24 C22.50C24 C42⋊3S3 C23.8D6 ...
C42⋊2C2 is a maximal quotient of
C42⋊5C4 C23.84C23
C23.D2p: C23.11D4 C23.8D6 C23.D10 C23.D14 C23.D22 C23.D26 ...
(C2×C4).D2p: C23.63C23 C24.C22 C23.Q8 C23.83C23 C42⋊3S3 C4⋊C4⋊S3 C42⋊2D5 C4⋊C4⋊D5 ...
Matrix representation of C42⋊2C2 ►in GL4(𝔽5) generated by
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(5))| [0,4,0,0,1,0,0,0,0,0,0,3,0,0,2,0],[2,0,0,0,0,2,0,0,0,0,0,4,0,0,1,0],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4] >;
C42⋊2C2 in GAP, Magma, Sage, TeX
C_4^2\rtimes_2C_2
% in TeX
G:=Group("C4^2:2C2"); // GroupNames label
G:=SmallGroup(32,33);
// by ID
G=gap.SmallGroup(32,33);
# by ID
G:=PCGroup([5,-2,2,2,-2,2,101,126,302,42]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^2=1,a*b=b*a,c*a*c=a*b^2,c*b*c=a^2*b^-1>;
// generators/relations
Export
Subgroup lattice of C42⋊2C2 in TeX
Character table of C42⋊2C2 in TeX