p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4≀C2, D4⋊2C4, Q8⋊2C4, C42⋊3C2, C4.17D4, C22.3D4, M4(2)⋊4C2, C4.3(C2×C4), C4○D4.1C2, C2.8(C22⋊C4), (C2×C4).16C22, 2-Sylow(GL(2,5)), SmallGroup(32,11)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4≀C2
G = < a,b,c | a4=b2=c4=1, bab=a-1, ac=ca, cbc-1=a-1b >
Character table of C4≀C2
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 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 | -i | 1 | i | i | -i | 1 | -i | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | -1 | i | 1 | -i | -i | i | 1 | i | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | -1 | -i | 1 | i | i | -i | -1 | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | -1 | i | 1 | -i | -i | i | -1 | -i | i | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 0 | 0 | -2i | 2i | -1-i | 0 | 1-i | -1+i | 1+i | 0 | 0 | 0 | complex faithful |
| ρ12 | 2 | -2 | 0 | 0 | 2i | -2i | -1+i | 0 | 1+i | -1-i | 1-i | 0 | 0 | 0 | complex faithful |
| ρ13 | 2 | -2 | 0 | 0 | -2i | 2i | 1+i | 0 | -1+i | 1-i | -1-i | 0 | 0 | 0 | complex faithful |
| ρ14 | 2 | -2 | 0 | 0 | 2i | -2i | 1-i | 0 | -1-i | 1+i | -1+i | 0 | 0 | 0 | complex faithful |
►On 8 points - transitive group 8T17
(1 2 3 4)(5 6 7 8)
(1 8)(2 7)(3 6)(4 5)
(1 3)(2 4)(5 6 7 8)
G:=sub<Sym(8)| (1,2,3,4)(5,6,7,8), (1,8)(2,7)(3,6)(4,5), (1,3)(2,4)(5,6,7,8)>;
G:=Group( (1,2,3,4)(5,6,7,8), (1,8)(2,7)(3,6)(4,5), (1,3)(2,4)(5,6,7,8) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8)], [(1,8),(2,7),(3,6),(4,5)], [(1,3),(2,4),(5,6,7,8)]])
G:=TransitiveGroup(8,17);
►On 16 points - transitive group 16T28
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 7)(9 10)(11 12)(14 16)
(1 8 11 16)(2 5 12 13)(3 6 9 14)(4 7 10 15)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,7)(9,10)(11,12)(14,16), (1,8,11,16)(2,5,12,13)(3,6,9,14)(4,7,10,15)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,7)(9,10)(11,12)(14,16), (1,8,11,16)(2,5,12,13)(3,6,9,14)(4,7,10,15) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,4),(2,3),(5,7),(9,10),(11,12),(14,16)], [(1,8,11,16),(2,5,12,13),(3,6,9,14),(4,7,10,15)]])
G:=TransitiveGroup(16,28);
►On 16 points - transitive group 16T42
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6)(2 5)(3 8)(4 7)(9 13)(10 16)(11 15)(12 14)
(1 15 3 13)(2 16 4 14)(5 9)(6 10)(7 11)(8 12)
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6)(2,5)(3,8)(4,7)(9,13)(10,16)(11,15)(12,14), (1,15,3,13)(2,16,4,14)(5,9)(6,10)(7,11)(8,12)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6)(2,5)(3,8)(4,7)(9,13)(10,16)(11,15)(12,14), (1,15,3,13)(2,16,4,14)(5,9)(6,10)(7,11)(8,12) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,6),(2,5),(3,8),(4,7),(9,13),(10,16),(11,15),(12,14)], [(1,15,3,13),(2,16,4,14),(5,9),(6,10),(7,11),(8,12)]])
G:=TransitiveGroup(16,42);
C4≀C2 is a maximal subgroup of
D4⋊4D4 D4.9D4 C42⋊S3 U2(𝔽3) D4⋊F5 C32⋊C4≀C2 Dic3≀C2 C32⋊6C4≀C2 C32⋊7C4≀C2 Dic26⋊C4
D4p⋊C4: C8○D8 C8.26D4 C42⋊4S3 D12⋊C4 D20⋊4C4 D20⋊7C4 Q8⋊2F5 Dic14⋊C4 ...
(C2×C2p).D4: C42⋊C22 D4.8D4 D4.10D4 Q8⋊3Dic3 D4⋊2Dic5 D4⋊2Dic7 C44.56D4 C52.56D4 ...
C4≀C2 is a maximal quotient of
D4⋊F5 Q8⋊2F5 C32⋊C4≀C2 Dic3≀C2 C32⋊6C4≀C2 C32⋊7C4≀C2 Dic26⋊C4 D52⋊C4
C4.D4p: D4⋊C8 C42⋊4S3 D20⋊4C4 Dic14⋊C4 D44⋊1C4 D52⋊4C4 ...
C22.D4p: C22.SD16 D12⋊C4 D20⋊7C4 D28⋊4C4 D44⋊4C4 D52⋊7C4 ...
(C2×C2p).D4: Q8⋊C8 C23.31D4 C42.C22 C42.2C22 C42⋊6C4 Q8⋊3Dic3 D4⋊2Dic5 D4⋊2Dic7 ...
| action | f(x) | Disc(f) |
|---|---|---|
| 8T17 | x8-4x7+14x5-8x4-12x3+7x2+2x-1 | 212·413 |
Matrix representation of C4≀C2 ►in GL2(𝔽5) generated by
| 3 | 0 |
| 0 | 2 |
| 0 | 2 |
| 3 | 0 |
| 2 | 0 |
| 0 | 1 |
G:=sub<GL(2,GF(5))| [3,0,0,2],[0,3,2,0],[2,0,0,1] >;
C4≀C2 in GAP, Magma, Sage, TeX
C_4\wr C_2
% in TeX
G:=Group("C4wrC2"); // GroupNames label
G:=SmallGroup(32,11);
// by ID
G=gap.SmallGroup(32,11);
# by ID
G:=PCGroup([5,-2,2,-2,2,-2,40,61,302,157,72,58]);
// Polycyclic
G:=Group<a,b,c|a^4=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^-1*b>;
// generators/relations
Export
Subgroup lattice of C4≀C2 in TeX
Character table of C4≀C2 in TeX