p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.287D4, C42.417C23, C4.592- 1+4, C8⋊Q8⋊13C2, D4⋊2Q8⋊9C2, D4⋊Q8⋊26C2, C4⋊C8.69C22, (C2×C8).69C23, C4⋊C4.174C23, (C2×C4).433C24, C23.299(C2×D4), (C22×C4).515D4, C4⋊Q8.316C22, C4.105(C8⋊C22), C8⋊C4.26C22, C4.Q8.37C22, C42.6C4⋊17C2, (C2×D4).179C23, (C4×D4).117C22, C22⋊C8.60C22, C2.D8.103C22, D4⋊C4.49C22, C4⋊1D4.173C22, C23.19D4⋊28C2, C4⋊D4.202C22, (C2×C42).894C22, C22.693(C22×D4), C2.64(D8⋊C22), (C22×C4).1098C23, C42.29C22⋊6C2, C42.C2.134C22, C42⋊C2.166C22, C23.37C23⋊22C2, C22.26C24.47C2, C2.81(C23.38C23), (C2×C4).557(C2×D4), C2.63(C2×C8⋊C22), SmallGroup(128,1967)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.287D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1, cbc-1=dbd=a2b, dcd=a2b2c3 >
Subgroups: 364 in 183 conjugacy classes, 86 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C4.4D4, C42.C2, C4⋊1D4, C4⋊Q8, C2×C4○D4, C42.6C4, D4⋊Q8, D4⋊2Q8, C23.19D4, C42.29C22, C8⋊Q8, C22.26C24, C23.37C23, C42.287D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8⋊C22, C22×D4, 2- 1+4, C23.38C23, C2×C8⋊C22, D8⋊C22, C42.287D4
Character table of C42.287D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 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 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -4i | 0 | 0 | 4i | 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 | 0 | 0 | 4i | 0 | 0 | -4i | 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 25 63 42)(2 30 64 47)(3 27 57 44)(4 32 58 41)(5 29 59 46)(6 26 60 43)(7 31 61 48)(8 28 62 45)(9 23 39 53)(10 20 40 50)(11 17 33 55)(12 22 34 52)(13 19 35 49)(14 24 36 54)(15 21 37 51)(16 18 38 56)
(1 17 5 21)(2 56 6 52)(3 19 7 23)(4 50 8 54)(9 44 13 48)(10 28 14 32)(11 46 15 42)(12 30 16 26)(18 60 22 64)(20 62 24 58)(25 33 29 37)(27 35 31 39)(34 47 38 43)(36 41 40 45)(49 61 53 57)(51 63 55 59)
(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 42)(2 32)(3 48)(4 30)(5 46)(6 28)(7 44)(8 26)(9 49)(10 18)(11 55)(12 24)(13 53)(14 22)(15 51)(16 20)(17 33)(19 39)(21 37)(23 35)(25 63)(27 61)(29 59)(31 57)(34 54)(36 52)(38 50)(40 56)(41 64)(43 62)(45 60)(47 58)
G:=sub<Sym(64)| (1,25,63,42)(2,30,64,47)(3,27,57,44)(4,32,58,41)(5,29,59,46)(6,26,60,43)(7,31,61,48)(8,28,62,45)(9,23,39,53)(10,20,40,50)(11,17,33,55)(12,22,34,52)(13,19,35,49)(14,24,36,54)(15,21,37,51)(16,18,38,56), (1,17,5,21)(2,56,6,52)(3,19,7,23)(4,50,8,54)(9,44,13,48)(10,28,14,32)(11,46,15,42)(12,30,16,26)(18,60,22,64)(20,62,24,58)(25,33,29,37)(27,35,31,39)(34,47,38,43)(36,41,40,45)(49,61,53,57)(51,63,55,59), (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), (1,42)(2,32)(3,48)(4,30)(5,46)(6,28)(7,44)(8,26)(9,49)(10,18)(11,55)(12,24)(13,53)(14,22)(15,51)(16,20)(17,33)(19,39)(21,37)(23,35)(25,63)(27,61)(29,59)(31,57)(34,54)(36,52)(38,50)(40,56)(41,64)(43,62)(45,60)(47,58)>;
G:=Group( (1,25,63,42)(2,30,64,47)(3,27,57,44)(4,32,58,41)(5,29,59,46)(6,26,60,43)(7,31,61,48)(8,28,62,45)(9,23,39,53)(10,20,40,50)(11,17,33,55)(12,22,34,52)(13,19,35,49)(14,24,36,54)(15,21,37,51)(16,18,38,56), (1,17,5,21)(2,56,6,52)(3,19,7,23)(4,50,8,54)(9,44,13,48)(10,28,14,32)(11,46,15,42)(12,30,16,26)(18,60,22,64)(20,62,24,58)(25,33,29,37)(27,35,31,39)(34,47,38,43)(36,41,40,45)(49,61,53,57)(51,63,55,59), (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), (1,42)(2,32)(3,48)(4,30)(5,46)(6,28)(7,44)(8,26)(9,49)(10,18)(11,55)(12,24)(13,53)(14,22)(15,51)(16,20)(17,33)(19,39)(21,37)(23,35)(25,63)(27,61)(29,59)(31,57)(34,54)(36,52)(38,50)(40,56)(41,64)(43,62)(45,60)(47,58) );
G=PermutationGroup([[(1,25,63,42),(2,30,64,47),(3,27,57,44),(4,32,58,41),(5,29,59,46),(6,26,60,43),(7,31,61,48),(8,28,62,45),(9,23,39,53),(10,20,40,50),(11,17,33,55),(12,22,34,52),(13,19,35,49),(14,24,36,54),(15,21,37,51),(16,18,38,56)], [(1,17,5,21),(2,56,6,52),(3,19,7,23),(4,50,8,54),(9,44,13,48),(10,28,14,32),(11,46,15,42),(12,30,16,26),(18,60,22,64),(20,62,24,58),(25,33,29,37),(27,35,31,39),(34,47,38,43),(36,41,40,45),(49,61,53,57),(51,63,55,59)], [(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)], [(1,42),(2,32),(3,48),(4,30),(5,46),(6,28),(7,44),(8,26),(9,49),(10,18),(11,55),(12,24),(13,53),(14,22),(15,51),(16,20),(17,33),(19,39),(21,37),(23,35),(25,63),(27,61),(29,59),(31,57),(34,54),(36,52),(38,50),(40,56),(41,64),(43,62),(45,60),(47,58)]])
Matrix representation of C42.287D4 ►in GL8(𝔽17)
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 1 | 2 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 15 | 2 |
| 0 | 0 | 0 | 0 | 0 | 3 | 2 | 2 |
| 0 | 0 | 0 | 0 | 2 | 15 | 14 | 0 |
| 0 | 0 | 0 | 0 | 15 | 15 | 0 | 14 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 16 | 15 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 16 | 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 | 4 |
| 15 | 13 | 0 | 6 | 0 | 0 | 0 | 0 |
| 15 | 15 | 6 | 6 | 0 | 0 | 0 | 0 |
| 15 | 15 | 15 | 15 | 0 | 0 | 0 | 0 |
| 15 | 0 | 4 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 3 |
| 0 | 0 | 0 | 0 | 15 | 2 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 15 | 2 |
| 0 | 0 | 0 | 0 | 0 | 14 | 15 | 15 |
| 16 | 16 | 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 15 | 2 |
| 0 | 0 | 0 | 0 | 0 | 14 | 15 | 15 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 3 |
| 0 | 0 | 0 | 0 | 15 | 2 | 3 | 0 |
G:=sub<GL(8,GF(17))| [0,16,1,16,0,0,0,0,0,16,0,16,0,0,0,0,16,1,0,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,3,0,2,15,0,0,0,0,0,3,15,15,0,0,0,0,15,2,14,0,0,0,0,0,2,2,0,14],[0,1,16,1,0,0,0,0,0,1,0,1,0,0,0,0,1,16,0,0,0,0,0,0,0,15,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[15,15,15,15,0,0,0,0,13,15,15,0,0,0,0,0,0,6,15,4,0,0,0,0,6,6,15,6,0,0,0,0,0,0,0,0,2,15,3,0,0,0,0,0,2,2,0,14,0,0,0,0,0,3,15,15,0,0,0,0,3,0,2,15],[16,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,1,16,0,0,0,0,0,0,2,0,0,1,0,0,0,0,0,0,0,0,3,0,2,15,0,0,0,0,0,14,2,2,0,0,0,0,15,15,0,3,0,0,0,0,2,15,3,0] >;
C42.287D4 in GAP, Magma, Sage, TeX
C_4^2._{287}D_4 % in TeX
G:=Group("C4^2.287D4"); // GroupNames label
G:=SmallGroup(128,1967);
// by ID
G=gap.SmallGroup(128,1967);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,891,100,675,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1,c*b*c^-1=d*b*d=a^2*b,d*c*d=a^2*b^2*c^3>;
// generators/relations
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