p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.6C4, C4.8M4(2), C42.68C22, C22.6M4(2), C4⋊C8⋊13C2, C8⋊C4⋊7C2, C22⋊C8.7C2, C4.49(C4○D4), (C2×C42).16C2, (C2×C8).47C22, C23.33(C2×C4), (C22×C4).12C4, C2.8(C2×M4(2)), (C2×C4).151C23, C22.45(C22×C4), (C22×C4).94C22, C2.11(C42⋊C2), (C2×C4).58(C2×C4), SmallGroup(64,113)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.6C4
G = < a,b,c | a4=b4=1, c4=a2, ab=ba, cac-1=a-1b2, cbc-1=a2b >
Character table of C42.6C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 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 | 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 | 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 | 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 | -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 | -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 | -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 | -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 | 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 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -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 | 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 | -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 | 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 | -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 | 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 | -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 | 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 | 2 | -2 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | -2i | 2i | 2i | 2 | -2 | 2i | 0 | -2i | 0 | 0 | 0 | 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 | -2 | 2 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2i | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | -2i | 2i | 2i | -2 | 2 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ23 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | -2i | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | -2 | 2 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ25 | 2 | -2 | 2 | -2 | -2 | 2 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ26 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 2i | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | 2i | -2i | -2i | 2 | -2 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ28 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2i | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
►On 32 points
(1 3 5 7)(2 26 6 30)(4 28 8 32)(9 24 13 20)(10 12 14 16)(11 18 15 22)(17 19 21 23)(25 27 29 31)
(1 21 27 12)(2 18 28 9)(3 23 29 14)(4 20 30 11)(5 17 31 16)(6 22 32 13)(7 19 25 10)(8 24 26 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
G:=sub<Sym(32)| (1,3,5,7)(2,26,6,30)(4,28,8,32)(9,24,13,20)(10,12,14,16)(11,18,15,22)(17,19,21,23)(25,27,29,31), (1,21,27,12)(2,18,28,9)(3,23,29,14)(4,20,30,11)(5,17,31,16)(6,22,32,13)(7,19,25,10)(8,24,26,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)>;
G:=Group( (1,3,5,7)(2,26,6,30)(4,28,8,32)(9,24,13,20)(10,12,14,16)(11,18,15,22)(17,19,21,23)(25,27,29,31), (1,21,27,12)(2,18,28,9)(3,23,29,14)(4,20,30,11)(5,17,31,16)(6,22,32,13)(7,19,25,10)(8,24,26,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32) );
G=PermutationGroup([[(1,3,5,7),(2,26,6,30),(4,28,8,32),(9,24,13,20),(10,12,14,16),(11,18,15,22),(17,19,21,23),(25,27,29,31)], [(1,21,27,12),(2,18,28,9),(3,23,29,14),(4,20,30,11),(5,17,31,16),(6,22,32,13),(7,19,25,10),(8,24,26,15)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)]])
C42.6C4 is a maximal subgroup of
C42.677C23 C42.261C23 C42.262C23 C42.678C23 C42.291C23 C42.292C23 D4⋊6M4(2) Q8⋊6M4(2) C23⋊3M4(2) D4⋊7M4(2) C42.300C23 C42.301C23 Q8.4M4(2) C42.696C23 C42.305C23 C42.698C23 D4⋊8M4(2) C42.307C23 C42.310C23 C42.15F5 C42.7F5
C42.D2p: C42.4Q8 C42.23D4 C42.6Q8 C42.25D4 C42.26D4 C42.9Q8 C42.370D4 C42.30D4 ...
(C2×C2p).M4(2): C42.302C23 Dic3.M4(2) Dic5.9M4(2) Dic5.13M4(2) C20.30M4(2) Dic7.M4(2) ...
C42.6C4 is a maximal quotient of
C43.C2 C23.32M4(2) C42⋊5C8 C8.5M4(2) C8.19M4(2) C42.15F5 C42.7F5 Dic5.13M4(2) C20.30M4(2)
C42.D2p: C42.378D4 C43.7C2 C42.425D4 C42⋊8C8 C42.182D6 C42.202D6 C42.270D6 C42.182D10 ...
(C2×C8).D2p: (C2×C8).Q8 C23.9M4(2) Dic3.M4(2) Dic5.9M4(2) Dic7.M4(2) ...
Matrix representation of C42.6C4 ►in GL4(𝔽17) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 13 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,13],[13,0,0,0,0,4,0,0,0,0,1,0,0,0,0,16],[0,4,0,0,1,0,0,0,0,0,0,4,0,0,1,0] >;
C42.6C4 in GAP, Magma, Sage, TeX
C_4^2._6C_4
% in TeX
G:=Group("C4^2.6C4"); // GroupNames label
G:=SmallGroup(64,113);
// by ID
G=gap.SmallGroup(64,113);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,50,88]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=1,c^4=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,c*b*c^-1=a^2*b>;
// generators/relations
Export
Subgroup lattice of C42.6C4 in TeX
Character table of C42.6C4 in TeX