p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.30C22, C4⋊Q8.8C2, (C2×C4).43D4, C8⋊C4.6C2, C4.18(C4○D4), C4⋊C4.21C22, (C2×C8).56C22, Q8⋊C4.7C2, C42.C2.3C2, (C2×C4).116C23, C22.112(C2×D4), (C2×Q8).23C22, C2.14(C4.4D4), C2.21(C8.C22), SmallGroup(64,172)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.30C22
G = < a,b,c,d | a4=b4=1, c2=b2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=a2b-1c >
Character table of C42.30C22
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | complex lifted from C4○D4 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | complex lifted from C4○D4 |
| ρ13 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | complex lifted from C4○D4 |
| ρ15 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ16 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►Regular action on 64 points
Generators in S64
(1 40 29 15)(2 37 30 12)(3 34 31 9)(4 39 32 14)(5 36 25 11)(6 33 26 16)(7 38 27 13)(8 35 28 10)(17 57 47 54)(18 62 48 51)(19 59 41 56)(20 64 42 53)(21 61 43 50)(22 58 44 55)(23 63 45 52)(24 60 46 49)
(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 59 5 63)(2 51 6 55)(3 57 7 61)(4 49 8 53)(9 47 13 43)(10 20 14 24)(11 45 15 41)(12 18 16 22)(17 38 21 34)(19 36 23 40)(25 52 29 56)(26 58 30 62)(27 50 31 54)(28 64 32 60)(33 44 37 48)(35 42 39 46)
(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)
G:=sub<Sym(64)| (1,40,29,15)(2,37,30,12)(3,34,31,9)(4,39,32,14)(5,36,25,11)(6,33,26,16)(7,38,27,13)(8,35,28,10)(17,57,47,54)(18,62,48,51)(19,59,41,56)(20,64,42,53)(21,61,43,50)(22,58,44,55)(23,63,45,52)(24,60,46,49), (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), (1,59,5,63)(2,51,6,55)(3,57,7,61)(4,49,8,53)(9,47,13,43)(10,20,14,24)(11,45,15,41)(12,18,16,22)(17,38,21,34)(19,36,23,40)(25,52,29,56)(26,58,30,62)(27,50,31,54)(28,64,32,60)(33,44,37,48)(35,42,39,46), (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)>;
G:=Group( (1,40,29,15)(2,37,30,12)(3,34,31,9)(4,39,32,14)(5,36,25,11)(6,33,26,16)(7,38,27,13)(8,35,28,10)(17,57,47,54)(18,62,48,51)(19,59,41,56)(20,64,42,53)(21,61,43,50)(22,58,44,55)(23,63,45,52)(24,60,46,49), (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), (1,59,5,63)(2,51,6,55)(3,57,7,61)(4,49,8,53)(9,47,13,43)(10,20,14,24)(11,45,15,41)(12,18,16,22)(17,38,21,34)(19,36,23,40)(25,52,29,56)(26,58,30,62)(27,50,31,54)(28,64,32,60)(33,44,37,48)(35,42,39,46), (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) );
G=PermutationGroup([[(1,40,29,15),(2,37,30,12),(3,34,31,9),(4,39,32,14),(5,36,25,11),(6,33,26,16),(7,38,27,13),(8,35,28,10),(17,57,47,54),(18,62,48,51),(19,59,41,56),(20,64,42,53),(21,61,43,50),(22,58,44,55),(23,63,45,52),(24,60,46,49)], [(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)], [(1,59,5,63),(2,51,6,55),(3,57,7,61),(4,49,8,53),(9,47,13,43),(10,20,14,24),(11,45,15,41),(12,18,16,22),(17,38,21,34),(19,36,23,40),(25,52,29,56),(26,58,30,62),(27,50,31,54),(28,64,32,60),(33,44,37,48),(35,42,39,46)], [(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)]])
C42.30C22 is a maximal subgroup of
C42.(C2×C4) C8⋊C4.C4
C42.D2p: C42.239D4 C42.241D4 C42.244D4 C42.256D4 C42.258D4 C42.276D4 C42.277D4 C42.288D4 ...
C4⋊C4.D2p: C42.8C23 C42.9C23 C42.10C23 C42.367C23 C42.386C23 C42.389C23 C42.407C23 C42.409C23 ...
C42.30C22 is a maximal quotient of
C42.24Q8 C2.(C8⋊D4) (C2×Q8).109D4 (C2×C4).28D8
C42.D2p: C42.111D4 C42.124D4 C42.14D6 C42.71D6 C42.77D6 C42.14D10 C42.71D10 C42.77D10 ...
C4⋊C4.D2p: C4⋊C4.85D4 C4⋊C4.95D4 (C2×Q8).36D6 C40⋊8C4.C2 C56⋊C4.C2 ...
Matrix representation of C42.30C22 ►in GL6(𝔽17)
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 16 | 0 | 0 |
| 0 | 0 | 16 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 7 |
| 0 | 0 | 0 | 0 | 7 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,13,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,7,16,0,0,0,0,16,10,0,0,0,0,0,0,1,7,0,0,0,0,7,16],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,16,0,0,0,0,1,0,0,0] >;
C42.30C22 in GAP, Magma, Sage, TeX
C_4^2._{30}C_2^2 % in TeX
G:=Group("C4^2.30C2^2"); // GroupNames label
G:=SmallGroup(64,172);
// by ID
G=gap.SmallGroup(64,172);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,295,362,332,50,963,117,1444,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^2=b^2,d^2=b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^-1*c>;
// generators/relations
Export
Subgroup lattice of C42.30C22 in TeX
Character table of C42.30C22 in TeX