p-group, metabelian, nilpotent (class 3), monomial
Aliases: C23.5Q8, M4(2).1C4, M4(2).10C22, C8.4(C2×C4), (C2×C4).7Q8, C4.75(C2×D4), C4.17(C4⋊C4), C8.C4⋊3C2, (C2×C4).129D4, C22.2(C2×Q8), (C2×C8).14C22, (C2×C4).72C23, C4.29(C22×C4), C22.11(C4⋊C4), (C2×M4(2)).2C2, (C22×C4).39C22, C2.16(C2×C4⋊C4), (C2×C4).27(C2×C4), SmallGroup(64,111)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for M4(2).C4
G = < a,b,c | a8=b2=1, c4=a4, bab=a5, cac-1=a-1, cbc-1=a4b >
Character table of M4(2).C4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 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 | -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 | -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 | 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 | 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 | -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 | -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 | 1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | -1 | -1 | i | 1 | 1 | -i | -i | i | i | i | -i | linear of order 4 |
| ρ10 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | 1 | -1 | -i | -1 | 1 | i | -i | i | -i | i | -i | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -1 | -1 | -i | 1 | 1 | i | i | -i | -i | -i | i | linear of order 4 |
| ρ12 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | 1 | -1 | i | -1 | 1 | -i | i | -i | i | -i | i | linear of order 4 |
| ρ13 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -1 | 1 | i | 1 | -1 | -i | i | i | -i | -i | -i | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | 1 | 1 | -i | -1 | -1 | i | i | i | i | -i | -i | linear of order 4 |
| ρ15 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | -1 | 1 | -i | 1 | -1 | i | -i | -i | i | i | i | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | 1 | 1 | i | -1 | -1 | -i | -i | -i | -i | i | i | linear of order 4 |
| ρ17 | 2 | 2 | -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 | -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 | -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 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 16 points - transitive group 16T103
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 5)(3 7)(10 14)(12 16)
(1 9 7 11 5 13 3 15)(2 16 8 10 6 12 4 14)
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(3,7)(10,14)(12,16), (1,9,7,11,5,13,3,15)(2,16,8,10,6,12,4,14)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(3,7)(10,14)(12,16), (1,9,7,11,5,13,3,15)(2,16,8,10,6,12,4,14) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,5),(3,7),(10,14),(12,16)], [(1,9,7,11,5,13,3,15),(2,16,8,10,6,12,4,14)]])
G:=TransitiveGroup(16,103);
M4(2).C4 is a maximal subgroup of
M4(2).29C23 M4(2).1F5
M4(2).D2p: M4(2).40D4 M4(2).41D4 C42.427D4 M4(2).27D4 M4(2).8D4 M4(2).9D4 C24.Q8 M4(2).15D4 ...
(C2×C8).D2p: (C2×C8).D4 (C2×C8).6D4 C24.11Q8 C42.10D4 C42.32Q8 C22⋊C4.Q8 M5(2).C22 C23.10SD16 ...
M4(2).C4 is a maximal quotient of
M4(2)⋊1C8 C8⋊1M4(2) C42.90D4 C42.91D4 C42.Q8 C24.7Q8 C8.6C42 C42.104D4 M4(2).1F5
M4(2).D2p: C24.10Q8 C42.430D4 M4(2).25D6 C23.8Dic6 M4(2).25D10 C23.Dic10 M4(2).25D14 C23.Dic14 ...
(C2×C8).D2p: C24.9Q8 C42.106D4 C23.9Dic6 M4(2).Dic5 M4(2).Dic7 ...
Matrix representation of M4(2).C4 ►in GL4(𝔽5) generated by
| 0 | 0 | 0 | 3 |
| 0 | 0 | 1 | 0 |
| 0 | 2 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 3 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(5))| [0,0,0,1,0,0,2,0,0,1,0,0,3,0,0,0],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[0,4,0,0,3,0,0,0,0,0,0,3,0,0,4,0] >;
M4(2).C4 in GAP, Magma, Sage, TeX
M_4(2).C_4
% in TeX
G:=Group("M4(2).C4"); // GroupNames label
G:=SmallGroup(64,111);
// by ID
G=gap.SmallGroup(64,111);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,55,332,963,117,88]);
// Polycyclic
G:=Group<a,b,c|a^8=b^2=1,c^4=a^4,b*a*b=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^4*b>;
// generators/relations
Export
Subgroup lattice of M4(2).C4 in TeX
Character table of M4(2).C4 in TeX