p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.2C22, C2.7C4≀C2, C4⋊C4.1C4, (C2×C4).97D4, C8⋊C4.3C2, C42.C2.1C2, C2.3(C4.10D4), C22.38(C22⋊C4), (C2×C4).11(C2×C4), SmallGroup(64,11)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.2C22
G = < a,b,c,d | a4=b4=1, c2=b, d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=a2b-1, dcd-1=a-1b2c >
Character table of C42.2C22
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 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 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | -1+i | 1+i | 0 | 0 | -1-i | 1-i | 0 | complex lifted from C4≀C2 |
| ρ12 | 2 | -2 | 2 | -2 | -2i | 0 | 0 | 2i | 0 | 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 | 2i | -2i | 0 | 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 | -2i | 2i | 0 | 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 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | -1-i | 1-i | 0 | 0 | -1+i | 1+i | 0 | complex lifted from C4≀C2 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 1-i | 0 | 0 | 1+i | -1+i | 0 | 0 | -1-i | complex lifted from C4≀C2 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 2i | -2i | 0 | 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 | 2i | 0 | 0 | -2i | 0 | 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 | symplectic lifted from C4.10D4, Schur index 2 |
►Regular action on 64 points
Generators in S64
(1 31 55 47)(2 28 56 44)(3 25 49 41)(4 30 50 46)(5 27 51 43)(6 32 52 48)(7 29 53 45)(8 26 54 42)(9 57 40 17)(10 62 33 22)(11 59 34 19)(12 64 35 24)(13 61 36 21)(14 58 37 18)(15 63 38 23)(16 60 39 20)
(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)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 63 5 59)(2 12 6 16)(3 21 7 17)(4 33 8 37)(9 41 13 45)(10 54 14 50)(11 31 15 27)(18 46 22 42)(19 55 23 51)(20 28 24 32)(25 36 29 40)(26 58 30 62)(34 47 38 43)(35 52 39 56)(44 64 48 60)(49 61 53 57)
G:=sub<Sym(64)| (1,31,55,47)(2,28,56,44)(3,25,49,41)(4,30,50,46)(5,27,51,43)(6,32,52,48)(7,29,53,45)(8,26,54,42)(9,57,40,17)(10,62,33,22)(11,59,34,19)(12,64,35,24)(13,61,36,21)(14,58,37,18)(15,63,38,23)(16,60,39,20), (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)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,63,5,59)(2,12,6,16)(3,21,7,17)(4,33,8,37)(9,41,13,45)(10,54,14,50)(11,31,15,27)(18,46,22,42)(19,55,23,51)(20,28,24,32)(25,36,29,40)(26,58,30,62)(34,47,38,43)(35,52,39,56)(44,64,48,60)(49,61,53,57)>;
G:=Group( (1,31,55,47)(2,28,56,44)(3,25,49,41)(4,30,50,46)(5,27,51,43)(6,32,52,48)(7,29,53,45)(8,26,54,42)(9,57,40,17)(10,62,33,22)(11,59,34,19)(12,64,35,24)(13,61,36,21)(14,58,37,18)(15,63,38,23)(16,60,39,20), (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)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,63,5,59)(2,12,6,16)(3,21,7,17)(4,33,8,37)(9,41,13,45)(10,54,14,50)(11,31,15,27)(18,46,22,42)(19,55,23,51)(20,28,24,32)(25,36,29,40)(26,58,30,62)(34,47,38,43)(35,52,39,56)(44,64,48,60)(49,61,53,57) );
G=PermutationGroup([[(1,31,55,47),(2,28,56,44),(3,25,49,41),(4,30,50,46),(5,27,51,43),(6,32,52,48),(7,29,53,45),(8,26,54,42),(9,57,40,17),(10,62,33,22),(11,59,34,19),(12,64,35,24),(13,61,36,21),(14,58,37,18),(15,63,38,23),(16,60,39,20)], [(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),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,63,5,59),(2,12,6,16),(3,21,7,17),(4,33,8,37),(9,41,13,45),(10,54,14,50),(11,31,15,27),(18,46,22,42),(19,55,23,51),(20,28,24,32),(25,36,29,40),(26,58,30,62),(34,47,38,43),(35,52,39,56),(44,64,48,60),(49,61,53,57)]])
C42.2C22 is a maximal subgroup of
C42.2C23 C42.3C23 C42.4C23 C42.6C23 C42.7C23 C42.8C23 C42.9C23 C42.10C23 C10.C4≀C2
C42.D2p: C42.66D4 C42.406D4 C42.408D4 C42.376D4 C42.68D4 C42.69D4 C42.71D4 C42.72D4 ...
C42.2C22 is a maximal quotient of
C4⋊C4⋊C8 C10.C4≀C2
C42.D2p: C42.7Q8 C42.2D6 C42.8D6 C42.2D10 C42.8D10 C42.2D14 C42.8D14 ...
Matrix representation of C42.2C22 ►in GL4(𝔽17) generated by
| 0 | 13 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 6 | 10 | 0 | 0 |
| 10 | 6 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 2 | 0 |
| 4 | 10 | 0 | 0 |
| 7 | 13 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(17))| [0,13,0,0,13,0,0,0,0,0,0,16,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[6,10,0,0,10,6,0,0,0,0,0,2,0,0,2,0],[4,7,0,0,10,13,0,0,0,0,5,12,0,0,12,12] >;
C42.2C22 in GAP, Magma, Sage, TeX
C_4^2._2C_2^2
% in TeX
G:=Group("C4^2.2C2^2"); // GroupNames label
G:=SmallGroup(64,11);
// by ID
G=gap.SmallGroup(64,11);
# by ID
G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,103,362,332,158,681,69]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^2=b,d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^-1*b^2*c>;
// generators/relations
Export
Subgroup lattice of C42.2C22 in TeX
Character table of C42.2C22 in TeX