p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.289D4, C42.419C23, C4.612- 1+4, C8⋊Q8⋊15C2, Q8⋊Q8⋊9C2, C4.Q16⋊26C2, C4⋊C8.71C22, (C2×C8).71C23, C4⋊C4.176C23, (C2×C4).435C24, (C22×C4).517D4, C23.300(C2×D4), C4⋊Q8.318C22, C8⋊C4.28C22, C4.Q8.39C22, C22⋊C8.62C22, C42.6C4.3C2, (C4×Q8).114C22, (C2×Q8).167C23, C2.D8.105C22, C4.101(C8.C22), (C2×C42).896C22, Q8⋊C4.49C22, C23.20D4.3C2, C22.695(C22×D4), C22⋊Q8.207C22, C2.66(D8⋊C22), (C22×C4).1100C23, C42.30C22⋊6C2, C42.C2.136C22, C42⋊C2.167C22, C23.37C23.42C2, C2.83(C23.38C23), (C2×C4).559(C2×D4), C2.63(C2×C8.C22), SmallGroup(128,1969)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.289D4
G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=a2c3 >
Subgroups: 268 in 161 conjugacy classes, 86 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C42⋊C2, C42⋊C2, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C42.6C4, Q8⋊Q8, C4.Q16, C23.20D4, C42.30C22, C8⋊Q8, C23.37C23, C42.289D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8.C22, C22×D4, 2- 1+4, C23.38C23, C2×C8.C22, D8⋊C22, C42.289D4
Character table of C42.289D4
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | -1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 4i | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -4i | 0 | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►On 64 points
Generators in S64
(1 43 20 30)(2 48 21 27)(3 45 22 32)(4 42 23 29)(5 47 24 26)(6 44 17 31)(7 41 18 28)(8 46 19 25)(9 56 64 38)(10 53 57 35)(11 50 58 40)(12 55 59 37)(13 52 60 34)(14 49 61 39)(15 54 62 36)(16 51 63 33)
(1 18 5 22)(2 8 6 4)(3 20 7 24)(9 11 13 15)(10 59 14 63)(12 61 16 57)(17 23 21 19)(25 31 29 27)(26 45 30 41)(28 47 32 43)(33 53 37 49)(34 36 38 40)(35 55 39 51)(42 48 46 44)(50 52 54 56)(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 54 5 50)(2 39 6 35)(3 52 7 56)(4 37 8 33)(9 45 13 41)(10 27 14 31)(11 43 15 47)(12 25 16 29)(17 53 21 49)(18 38 22 34)(19 51 23 55)(20 36 24 40)(26 58 30 62)(28 64 32 60)(42 59 46 63)(44 57 48 61)
G:=sub<Sym(64)| (1,43,20,30)(2,48,21,27)(3,45,22,32)(4,42,23,29)(5,47,24,26)(6,44,17,31)(7,41,18,28)(8,46,19,25)(9,56,64,38)(10,53,57,35)(11,50,58,40)(12,55,59,37)(13,52,60,34)(14,49,61,39)(15,54,62,36)(16,51,63,33), (1,18,5,22)(2,8,6,4)(3,20,7,24)(9,11,13,15)(10,59,14,63)(12,61,16,57)(17,23,21,19)(25,31,29,27)(26,45,30,41)(28,47,32,43)(33,53,37,49)(34,36,38,40)(35,55,39,51)(42,48,46,44)(50,52,54,56)(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), (1,54,5,50)(2,39,6,35)(3,52,7,56)(4,37,8,33)(9,45,13,41)(10,27,14,31)(11,43,15,47)(12,25,16,29)(17,53,21,49)(18,38,22,34)(19,51,23,55)(20,36,24,40)(26,58,30,62)(28,64,32,60)(42,59,46,63)(44,57,48,61)>;
G:=Group( (1,43,20,30)(2,48,21,27)(3,45,22,32)(4,42,23,29)(5,47,24,26)(6,44,17,31)(7,41,18,28)(8,46,19,25)(9,56,64,38)(10,53,57,35)(11,50,58,40)(12,55,59,37)(13,52,60,34)(14,49,61,39)(15,54,62,36)(16,51,63,33), (1,18,5,22)(2,8,6,4)(3,20,7,24)(9,11,13,15)(10,59,14,63)(12,61,16,57)(17,23,21,19)(25,31,29,27)(26,45,30,41)(28,47,32,43)(33,53,37,49)(34,36,38,40)(35,55,39,51)(42,48,46,44)(50,52,54,56)(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), (1,54,5,50)(2,39,6,35)(3,52,7,56)(4,37,8,33)(9,45,13,41)(10,27,14,31)(11,43,15,47)(12,25,16,29)(17,53,21,49)(18,38,22,34)(19,51,23,55)(20,36,24,40)(26,58,30,62)(28,64,32,60)(42,59,46,63)(44,57,48,61) );
G=PermutationGroup([[(1,43,20,30),(2,48,21,27),(3,45,22,32),(4,42,23,29),(5,47,24,26),(6,44,17,31),(7,41,18,28),(8,46,19,25),(9,56,64,38),(10,53,57,35),(11,50,58,40),(12,55,59,37),(13,52,60,34),(14,49,61,39),(15,54,62,36),(16,51,63,33)], [(1,18,5,22),(2,8,6,4),(3,20,7,24),(9,11,13,15),(10,59,14,63),(12,61,16,57),(17,23,21,19),(25,31,29,27),(26,45,30,41),(28,47,32,43),(33,53,37,49),(34,36,38,40),(35,55,39,51),(42,48,46,44),(50,52,54,56),(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)], [(1,54,5,50),(2,39,6,35),(3,52,7,56),(4,37,8,33),(9,45,13,41),(10,27,14,31),(11,43,15,47),(12,25,16,29),(17,53,21,49),(18,38,22,34),(19,51,23,55),(20,36,24,40),(26,58,30,62),(28,64,32,60),(42,59,46,63),(44,57,48,61)]])
Matrix representation of C42.289D4 ►in GL8(𝔽17)
| 0 | 16 | 15 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 13 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 4 |
| 7 | 11 | 5 | 0 | 0 | 0 | 0 | 0 |
| 10 | 6 | 0 | 3 | 0 | 0 | 0 | 0 |
| 15 | 2 | 10 | 7 | 0 | 0 | 0 | 0 |
| 2 | 15 | 6 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 8 |
| 0 | 0 | 0 | 0 | 15 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 2 | 0 | 0 |
| 10 | 6 | 0 | 3 | 0 | 0 | 0 | 0 |
| 7 | 11 | 5 | 0 | 0 | 0 | 0 | 0 |
| 8 | 8 | 6 | 6 | 0 | 0 | 0 | 0 |
| 9 | 9 | 7 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 8 |
G:=sub<GL(8,GF(17))| [0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,0,0,1,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,4,15,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,15,0,0,0,0,0,0,0,13],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,4],[7,10,15,2,0,0,0,0,11,6,2,15,0,0,0,0,5,0,10,6,0,0,0,0,0,3,7,11,0,0,0,0,0,0,0,0,0,0,15,7,0,0,0,0,0,0,9,2,0,0,0,0,9,10,0,0,0,0,0,0,2,8,0,0],[10,7,8,9,0,0,0,0,6,11,8,9,0,0,0,0,0,5,6,7,0,0,0,0,3,0,6,7,0,0,0,0,0,0,0,0,2,10,0,0,0,0,0,0,8,15,0,0,0,0,0,0,0,0,9,10,0,0,0,0,0,0,2,8] >;
C42.289D4 in GAP, Magma, Sage, TeX
C_4^2._{289}D_4 % in TeX
G:=Group("C4^2.289D4"); // GroupNames label
G:=SmallGroup(128,1969);
// by ID
G=gap.SmallGroup(128,1969);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,891,100,675,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^3>;
// generators/relations
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