p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.1C22, C2.6C4≀C2, C8⋊C4⋊6C2, (C2×D4).1C4, (C2×C4).96D4, (C2×Q8).1C4, C4.4D4.1C2, C2.3(C4.D4), C22.37(C22⋊C4), (C2×C4).10(C2×C4), SmallGroup(64,10)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.C22
G = < a,b,c,d | a4=b4=d2=1, c2=b, ab=ba, cac-1=ab2, dad=a-1, bc=cb, dbd=a2b-1, dcd=a-1b2c >
Character table of C42.C22
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 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 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 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 | i | i | -i | -i | i | -i | i | -i | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | i | -i | i | i | -i | -i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -i | -i | i | i | -i | i | -i | i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -i | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 1-i | -1-i | 0 | 0 | 1+i | -1+i | 0 | complex lifted from C4≀C2 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 1+i | -1+i | 0 | 0 | 1-i | -1-i | 0 | complex lifted from C4≀C2 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 1+i | 0 | 0 | -1+i | -1-i | 0 | 0 | 1-i | complex lifted from C4≀C2 |
| ρ14 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -1+i | 0 | 0 | 1+i | 1-i | 0 | 0 | -1-i | complex lifted from C4≀C2 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 1-i | 0 | 0 | -1-i | -1+i | 0 | 0 | 1+i | complex lifted from C4≀C2 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | -1+i | 1+i | 0 | 0 | -1-i | 1-i | 0 | complex lifted from C4≀C2 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | -1-i | 0 | 0 | 1-i | 1+i | 0 | 0 | -1+i | complex lifted from C4≀C2 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | -1-i | 1-i | 0 | 0 | -1+i | 1+i | 0 | complex lifted from C4≀C2 |
| ρ19 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
►On 32 points
(1 22 31 15)(2 19 32 12)(3 24 25 9)(4 21 26 14)(5 18 27 11)(6 23 28 16)(7 20 29 13)(8 17 30 10)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(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)
(2 12)(3 29)(4 17)(6 16)(7 25)(8 21)(9 13)(10 26)(11 18)(14 30)(15 22)(19 32)(20 24)(23 28)
G:=sub<Sym(32)| (1,22,31,15)(2,19,32,12)(3,24,25,9)(4,21,26,14)(5,18,27,11)(6,23,28,16)(7,20,29,13)(8,17,30,10), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (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), (2,12)(3,29)(4,17)(6,16)(7,25)(8,21)(9,13)(10,26)(11,18)(14,30)(15,22)(19,32)(20,24)(23,28)>;
G:=Group( (1,22,31,15)(2,19,32,12)(3,24,25,9)(4,21,26,14)(5,18,27,11)(6,23,28,16)(7,20,29,13)(8,17,30,10), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (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), (2,12)(3,29)(4,17)(6,16)(7,25)(8,21)(9,13)(10,26)(11,18)(14,30)(15,22)(19,32)(20,24)(23,28) );
G=PermutationGroup([[(1,22,31,15),(2,19,32,12),(3,24,25,9),(4,21,26,14),(5,18,27,11),(6,23,28,16),(7,20,29,13),(8,17,30,10)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32)], [(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)], [(2,12),(3,29),(4,17),(6,16),(7,25),(8,21),(9,13),(10,26),(11,18),(14,30),(15,22),(19,32),(20,24),(23,28)]])
C42.C22 is a maximal subgroup of
C42.C23 C42.2C23 C42.3C23 C42.5C23 C42.6C23 C42.7C23 C42.8C23 C42.10C23 (C2×D4).F5 (C2×Q8).F5
C42.D2p: C42.2D4 C42.3D4 C42.66D4 C42.405D4 C42.407D4 C42.376D4 C42.67D4 C42.69D4 ...
C42.C22 is a maximal quotient of
(C2×C4).98D8 (C2×Q8)⋊C8 C4.C4≀C2 C42.(C2×C4) (C2×D4).F5 (C2×Q8).F5
C42.D2p: C42.7Q8 C42.D6 C42.7D6 C42.D10 C42.7D10 C42.D14 C42.7D14 ...
Matrix representation of C42.C22 ►in GL4(𝔽17) generated by
| 0 | 13 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 |
| 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 11 | 7 | 0 | 0 |
| 7 | 11 | 0 | 0 |
| 0 | 0 | 14 | 14 |
| 0 | 0 | 10 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 16 |
G:=sub<GL(4,GF(17))| [0,13,0,0,13,0,0,0,0,0,16,1,0,0,15,1],[0,1,0,0,1,0,0,0,0,0,13,0,0,0,0,13],[11,7,0,0,7,11,0,0,0,0,14,10,0,0,14,3],[1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16] >;
C42.C22 in GAP, Magma, Sage, TeX
C_4^2.C_2^2
% in TeX
G:=Group("C4^2.C2^2"); // GroupNames label
G:=SmallGroup(64,10);
// by ID
G=gap.SmallGroup(64,10);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,362,332,158,681,69]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^2=b,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1,b*c=c*b,d*b*d=a^2*b^-1,d*c*d=a^-1*b^2*c>;
// generators/relations
Export
Subgroup lattice of C42.C22 in TeX
Character table of C42.C22 in TeX