p-group, metabelian, nilpotent (class 4), monomial
Aliases: C42.3C4, (C2×C4).4D4, C4⋊Q8.3C2, (C2×Q8).3C4, C4.10D4.C2, (C2×Q8).2C22, C2.11(C23⋊C4), C22.14(C22⋊C4), (C2×C4).4(C2×C4), SmallGroup(64,37)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.3C4
G = < a,b,c | a4=b4=1, c4=b2, ab=ba, cac-1=a-1b, cbc-1=a2b >
Character table of C42.3C4
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -i | i | i | -i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ12 | 4 | -4 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ13 | 4 | -4 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 16 points - transitive group 16T139
(1 5)(2 16 6 12)(3 7)(4 14 8 10)(9 13)(11 15)
(1 11 5 15)(2 12 6 16)(3 9 7 13)(4 10 8 14)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
G:=sub<Sym(16)| (1,5)(2,16,6,12)(3,7)(4,14,8,10)(9,13)(11,15), (1,11,5,15)(2,12,6,16)(3,9,7,13)(4,10,8,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;
G:=Group( (1,5)(2,16,6,12)(3,7)(4,14,8,10)(9,13)(11,15), (1,11,5,15)(2,12,6,16)(3,9,7,13)(4,10,8,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );
G=PermutationGroup([[(1,5),(2,16,6,12),(3,7),(4,14,8,10),(9,13),(11,15)], [(1,11,5,15),(2,12,6,16),(3,9,7,13),(4,10,8,14)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)]])
G:=TransitiveGroup(16,139);
C42.3C4 is a maximal subgroup of
(C2×D4).135D4 (C2×D4).137D4 C42.F5 (Q8×C10).C4
(C2×Q8).D2p: C42.4D4 (C4×C8).C4 (C2×Q8).D4 C8⋊C4.C4 C4⋊Q8.C4 C42.16D4 C42.17D4 Q8≀C2 ...
C42.3C4 is a maximal quotient of
(C2×C42).C4 C42⋊3C8 C2.7C2≀C4 C42.F5 (Q8×C10).C4
(C2×C4).D4p: (C2×C4).D8 (C2×C12).D4 (C2×Q8).D10 (C2×Q8).D14 ...
(C2×Q8).D2p: (C2×Q8).Q8 C42.3Dic3 C42.3Dic5 C42.3Dic7 ...
Matrix representation of C42.3C4 ►in GL4(𝔽3) generated by
| 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 0 | 0 | 2 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 2 |
| 0 | 0 | 2 | 0 |
| 1 | 0 | 0 | 2 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 0 |
G:=sub<GL(4,GF(3))| [1,0,0,0,0,0,1,0,0,2,0,0,0,0,0,1],[1,0,0,1,0,0,1,0,0,2,0,0,1,0,0,2],[0,1,0,0,0,0,0,1,2,0,0,1,0,2,1,0] >;
C42.3C4 in GAP, Magma, Sage, TeX
C_4^2._3C_4
% in TeX
G:=Group("C4^2.3C4"); // GroupNames label
G:=SmallGroup(64,37);
// by ID
G=gap.SmallGroup(64,37);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,199,362,332,158,681,255,117,1444]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=1,c^4=b^2,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^2*b>;
// generators/relations
Export
Subgroup lattice of C42.3C4 in TeX
Character table of C42.3C4 in TeX