p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.286D4, C42.416C23, C4.582- 1+4, C8⋊Q8⋊12C2, D4.Q8⋊23C2, C4⋊C8.68C22, (C2×C8).68C23, C4⋊C4.173C23, (C2×C4).432C24, C23.702(C2×D4), (C22×C4).514D4, C4⋊Q8.315C22, C8⋊C4.25C22, C42.6C4⋊16C2, C4.Q8.36C22, (C2×D4).178C23, (C4×D4).116C22, C22.D8⋊23C2, C22⋊C8.59C22, C2.D8.102C22, D4⋊C4.48C22, C4⋊D4.201C22, C23.46D4⋊12C2, C4⋊1D4.172C22, C22.52(C8⋊C22), (C2×C42).893C22, C22.692(C22×D4), C2.63(D8⋊C22), (C22×C4).1097C23, C42.29C22⋊5C2, C42.C2.133C22, C22.26C24.46C2, C2.80(C23.38C23), (C2×C4).556(C2×D4), C2.62(C2×C8⋊C22), (C2×C42.C2)⋊37C2, (C2×C4⋊C4).650C22, SmallGroup(128,1966)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.286D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1, cbc-1=a2b-1, dbd=a2b, dcd=a2c3 >
Subgroups: 364 in 184 conjugacy classes, 86 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, 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, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C42.C2, C42.C2, C4⋊1D4, C4⋊Q8, C2×C4○D4, C42.6C4, D4.Q8, C22.D8, C23.46D4, C42.29C22, C8⋊Q8, C2×C42.C2, C22.26C24, C42.286D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8⋊C22, C22×D4, 2- 1+4, C23.38C23, C2×C8⋊C22, D8⋊C22, C42.286D4
Character table of C42.286D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 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 | -2 | 0 | 0 | -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 | 2 | 0 | 0 | 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 | 2 | 0 | 0 | -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 | -2 | 0 | 0 | 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 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ22 | 4 | -4 | -4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 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 | 0 | 0 | 4 | -4 | 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 | 0 | 0 | -4 | 4 | 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 | 4i | -4i | 0 | 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 | 0 | -4i | 4i | 0 | 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 46 25 59)(2 43 26 64)(3 48 27 61)(4 45 28 58)(5 42 29 63)(6 47 30 60)(7 44 31 57)(8 41 32 62)(9 56 34 20)(10 53 35 17)(11 50 36 22)(12 55 37 19)(13 52 38 24)(14 49 39 21)(15 54 40 18)(16 51 33 23)
(1 11 5 15)(2 33 6 37)(3 13 7 9)(4 35 8 39)(10 32 14 28)(12 26 16 30)(17 41 21 45)(18 59 22 63)(19 43 23 47)(20 61 24 57)(25 36 29 40)(27 38 31 34)(42 54 46 50)(44 56 48 52)(49 58 53 62)(51 60 55 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)
(2 28)(3 7)(4 26)(6 32)(8 30)(9 38)(10 16)(11 36)(12 14)(13 34)(15 40)(17 51)(19 49)(20 24)(21 55)(23 53)(27 31)(33 35)(37 39)(41 47)(42 63)(43 45)(44 61)(46 59)(48 57)(52 56)(58 64)(60 62)
G:=sub<Sym(64)| (1,46,25,59)(2,43,26,64)(3,48,27,61)(4,45,28,58)(5,42,29,63)(6,47,30,60)(7,44,31,57)(8,41,32,62)(9,56,34,20)(10,53,35,17)(11,50,36,22)(12,55,37,19)(13,52,38,24)(14,49,39,21)(15,54,40,18)(16,51,33,23), (1,11,5,15)(2,33,6,37)(3,13,7,9)(4,35,8,39)(10,32,14,28)(12,26,16,30)(17,41,21,45)(18,59,22,63)(19,43,23,47)(20,61,24,57)(25,36,29,40)(27,38,31,34)(42,54,46,50)(44,56,48,52)(49,58,53,62)(51,60,55,64), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,38)(10,16)(11,36)(12,14)(13,34)(15,40)(17,51)(19,49)(20,24)(21,55)(23,53)(27,31)(33,35)(37,39)(41,47)(42,63)(43,45)(44,61)(46,59)(48,57)(52,56)(58,64)(60,62)>;
G:=Group( (1,46,25,59)(2,43,26,64)(3,48,27,61)(4,45,28,58)(5,42,29,63)(6,47,30,60)(7,44,31,57)(8,41,32,62)(9,56,34,20)(10,53,35,17)(11,50,36,22)(12,55,37,19)(13,52,38,24)(14,49,39,21)(15,54,40,18)(16,51,33,23), (1,11,5,15)(2,33,6,37)(3,13,7,9)(4,35,8,39)(10,32,14,28)(12,26,16,30)(17,41,21,45)(18,59,22,63)(19,43,23,47)(20,61,24,57)(25,36,29,40)(27,38,31,34)(42,54,46,50)(44,56,48,52)(49,58,53,62)(51,60,55,64), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,38)(10,16)(11,36)(12,14)(13,34)(15,40)(17,51)(19,49)(20,24)(21,55)(23,53)(27,31)(33,35)(37,39)(41,47)(42,63)(43,45)(44,61)(46,59)(48,57)(52,56)(58,64)(60,62) );
G=PermutationGroup([[(1,46,25,59),(2,43,26,64),(3,48,27,61),(4,45,28,58),(5,42,29,63),(6,47,30,60),(7,44,31,57),(8,41,32,62),(9,56,34,20),(10,53,35,17),(11,50,36,22),(12,55,37,19),(13,52,38,24),(14,49,39,21),(15,54,40,18),(16,51,33,23)], [(1,11,5,15),(2,33,6,37),(3,13,7,9),(4,35,8,39),(10,32,14,28),(12,26,16,30),(17,41,21,45),(18,59,22,63),(19,43,23,47),(20,61,24,57),(25,36,29,40),(27,38,31,34),(42,54,46,50),(44,56,48,52),(49,58,53,62),(51,60,55,64)], [(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)], [(2,28),(3,7),(4,26),(6,32),(8,30),(9,38),(10,16),(11,36),(12,14),(13,34),(15,40),(17,51),(19,49),(20,24),(21,55),(23,53),(27,31),(33,35),(37,39),(41,47),(42,63),(43,45),(44,61),(46,59),(48,57),(52,56),(58,64),(60,62)]])
Matrix representation of C42.286D4 ►in GL8(𝔽17)
| 2 | 16 | 13 | 13 | 0 | 0 | 0 | 0 |
| 5 | 2 | 0 | 13 | 0 | 0 | 0 | 0 |
| 12 | 16 | 3 | 6 | 0 | 0 | 0 | 0 |
| 5 | 0 | 12 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 16 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 1 | 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(17))| [2,5,12,5,0,0,0,0,16,2,16,0,0,0,0,0,13,0,3,12,0,0,0,0,13,13,6,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,1,1,0,0,0,0,0,16,0,1,16,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,16,1,0,0,0,0,0,0,15,0,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,1,0,0,0,0,0,0,16,1,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C42.286D4 in GAP, Magma, Sage, TeX
C_4^2._{286}D_4 % in TeX
G:=Group("C4^2.286D4"); // GroupNames label
G:=SmallGroup(128,1966);
// by ID
G=gap.SmallGroup(128,1966);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,891,100,675,4037,1027,124]);
// 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=a^2*b^-1,d*b*d=a^2*b,d*c*d=a^2*c^3>;
// generators/relations
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