direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×M4(2), C8⋊4C22, C4○M4(2), C23.3C4, C4.11C23, (C2×C8)⋊6C2, (C2×C4).6C4, C4.10(C2×C4), (C22×C4).6C2, C2.6(C22×C4), (C2×C4).24C22, C22.11(C2×C4), SmallGroup(32,37)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×M4(2)
G = < a,b,c | a2=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Character table of C2×M4(2)
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 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 | -1 | -1 | i | -i | -i | -i | i | i | i | -i | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | -i | -i | i | i | -i | i | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -i | i | i | i | -i | -i | -i | i | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | i | i | -i | -i | i | -i | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -i | i | i | -i | -i | i | i | -i | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | i | -i | i | -i | -i | i | -i | i | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | i | -i | -i | i | i | -i | -i | i | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | -i | i | i | -i | i | -i | linear of order 4 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
►On 16 points - transitive group 16T15
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 14)(2 11)(3 16)(4 13)(5 10)(6 15)(7 12)(8 9)
G:=sub<Sym(16)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,11)(3,16)(4,13)(5,10)(6,15)(7,12)(8,9)>;
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,14)(2,11)(3,16)(4,13)(5,10)(6,15)(7,12)(8,9) );
G=PermutationGroup([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,14),(2,11),(3,16),(4,13),(5,10),(6,15),(7,12),(8,9)]])
G:=TransitiveGroup(16,15);
C2×M4(2) is a maximal subgroup of
C4.9C42 C4.10C42 C42⋊6C4 C4.C42 C22.C42 M4(2)⋊4C4 C23.C8 C8○2M4(2) C24.4C4 (C22×C8)⋊C2 M4(2).8C22 C23.36D4 C23.37D4 C23.38D4 C42⋊C22 C4⋊M4(2) C42.6C22 M4(2)⋊C4 M4(2).C4 C8⋊9D4 C8⋊6D4 C8⋊D4 C8⋊2D4 C8.D4 Q8○M4(2) D8⋊C22 D5⋊M4(2) C3⋊S3⋊M4(2) D13⋊M4(2)
C2×M4(2) is a maximal quotient of
C24.4C4 C4⋊M4(2) C42.12C4 C42.6C4 C8⋊9D4 C8⋊6D4 C8⋊4Q8 D5⋊M4(2) C3⋊S3⋊M4(2) D13⋊M4(2)
Matrix representation of C2×M4(2) ►in GL3(𝔽17) generated by
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 4 | 0 |
| 16 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,0,4,0,1,0],[16,0,0,0,1,0,0,0,16] >;
C2×M4(2) in GAP, Magma, Sage, TeX
C_2\times M_4(2)
% in TeX
G:=Group("C2xM4(2)"); // GroupNames label
G:=SmallGroup(32,37);
// by ID
G=gap.SmallGroup(32,37);
# by ID
G:=PCGroup([5,-2,2,2,-2,-2,40,181,58]);
// Polycyclic
G:=Group<a,b,c|a^2=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations
Export
Subgroup lattice of C2×M4(2) in TeX
Character table of C2×M4(2) in TeX