p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C42.291D4, C42.421C23, C4.632- 1+4, C8⋊Q8⋊17C2, Q8⋊Q8⋊10C2, D4⋊Q8⋊27C2, C4⋊C8.73C22, (C2×C8).73C23, C4⋊C4.178C23, (C2×C4).437C24, C23.705(C2×D4), (C22×C4).519D4, C4⋊Q8.320C22, C4.127(C8⋊C22), C8⋊C4.30C22, C4.Q8.41C22, C42.6C4⋊19C2, (C4×D4).119C22, (C2×D4).181C23, C22⋊C8.64C22, (C2×Q8).169C23, (C4×Q8).116C22, C2.D8.107C22, D4⋊C4.51C22, C23.48D4⋊23C2, C4⋊D4.204C22, (C2×C42).898C22, Q8⋊C4.51C22, C23.46D4.2C2, C22.697(C22×D4), C22⋊Q8.209C22, C42.28C22⋊8C2, (C22×C4).1102C23, C4.4D4.161C22, C22.22(C8.C22), C42.C2.138C22, C23.36C23.28C2, C2.85(C23.38C23), (C2×C4⋊Q8)⋊45C2, (C2×C4).561(C2×D4), C2.65(C2×C8⋊C22), C2.65(C2×C8.C22), (C2×C4⋊C4).653C22, SmallGroup(128,1971)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.291D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=a2c3 >
Subgroups: 340 in 181 conjugacy classes, 88 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C22×Q8, C42.6C4, D4⋊Q8, Q8⋊Q8, C23.46D4, C23.48D4, C42.28C22, C8⋊Q8, C23.36C23, C2×C4⋊Q8, C42.291D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8⋊C22, C8.C22, C22×D4, 2- 1+4, C23.38C23, C2×C8⋊C22, C2×C8.C22, C42.291D4
Character table of C42.291D4
| 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 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 4 | -4 | 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 | 0 | 0 | 0 | -4 | 4 | 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 | 4 | -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 | 4 | -4 | 0 | 0 | 0 | 0 | 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 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -4 | 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 |
►On 64 points
(1 44 29 63)(2 41 30 60)(3 46 31 57)(4 43 32 62)(5 48 25 59)(6 45 26 64)(7 42 27 61)(8 47 28 58)(9 54 36 23)(10 51 37 20)(11 56 38 17)(12 53 39 22)(13 50 40 19)(14 55 33 24)(15 52 34 21)(16 49 35 18)
(1 9 5 13)(2 37 6 33)(3 11 7 15)(4 39 8 35)(10 26 14 30)(12 28 16 32)(17 61 21 57)(18 43 22 47)(19 63 23 59)(20 45 24 41)(25 40 29 36)(27 34 31 38)(42 52 46 56)(44 54 48 50)(49 62 53 58)(51 64 55 60)
(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 32)(3 7)(4 30)(6 28)(8 26)(9 40)(10 16)(11 38)(12 14)(13 36)(15 34)(17 21)(18 55)(20 53)(22 51)(24 49)(27 31)(33 39)(35 37)(41 47)(42 61)(43 45)(44 59)(46 57)(48 63)(52 56)(58 60)(62 64)
G:=sub<Sym(64)| (1,44,29,63)(2,41,30,60)(3,46,31,57)(4,43,32,62)(5,48,25,59)(6,45,26,64)(7,42,27,61)(8,47,28,58)(9,54,36,23)(10,51,37,20)(11,56,38,17)(12,53,39,22)(13,50,40,19)(14,55,33,24)(15,52,34,21)(16,49,35,18), (1,9,5,13)(2,37,6,33)(3,11,7,15)(4,39,8,35)(10,26,14,30)(12,28,16,32)(17,61,21,57)(18,43,22,47)(19,63,23,59)(20,45,24,41)(25,40,29,36)(27,34,31,38)(42,52,46,56)(44,54,48,50)(49,62,53,58)(51,64,55,60), (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), (2,32)(3,7)(4,30)(6,28)(8,26)(9,40)(10,16)(11,38)(12,14)(13,36)(15,34)(17,21)(18,55)(20,53)(22,51)(24,49)(27,31)(33,39)(35,37)(41,47)(42,61)(43,45)(44,59)(46,57)(48,63)(52,56)(58,60)(62,64)>;
G:=Group( (1,44,29,63)(2,41,30,60)(3,46,31,57)(4,43,32,62)(5,48,25,59)(6,45,26,64)(7,42,27,61)(8,47,28,58)(9,54,36,23)(10,51,37,20)(11,56,38,17)(12,53,39,22)(13,50,40,19)(14,55,33,24)(15,52,34,21)(16,49,35,18), (1,9,5,13)(2,37,6,33)(3,11,7,15)(4,39,8,35)(10,26,14,30)(12,28,16,32)(17,61,21,57)(18,43,22,47)(19,63,23,59)(20,45,24,41)(25,40,29,36)(27,34,31,38)(42,52,46,56)(44,54,48,50)(49,62,53,58)(51,64,55,60), (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), (2,32)(3,7)(4,30)(6,28)(8,26)(9,40)(10,16)(11,38)(12,14)(13,36)(15,34)(17,21)(18,55)(20,53)(22,51)(24,49)(27,31)(33,39)(35,37)(41,47)(42,61)(43,45)(44,59)(46,57)(48,63)(52,56)(58,60)(62,64) );
G=PermutationGroup([[(1,44,29,63),(2,41,30,60),(3,46,31,57),(4,43,32,62),(5,48,25,59),(6,45,26,64),(7,42,27,61),(8,47,28,58),(9,54,36,23),(10,51,37,20),(11,56,38,17),(12,53,39,22),(13,50,40,19),(14,55,33,24),(15,52,34,21),(16,49,35,18)], [(1,9,5,13),(2,37,6,33),(3,11,7,15),(4,39,8,35),(10,26,14,30),(12,28,16,32),(17,61,21,57),(18,43,22,47),(19,63,23,59),(20,45,24,41),(25,40,29,36),(27,34,31,38),(42,52,46,56),(44,54,48,50),(49,62,53,58),(51,64,55,60)], [(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)], [(2,32),(3,7),(4,30),(6,28),(8,26),(9,40),(10,16),(11,38),(12,14),(13,36),(15,34),(17,21),(18,55),(20,53),(22,51),(24,49),(27,31),(33,39),(35,37),(41,47),(42,61),(43,45),(44,59),(46,57),(48,63),(52,56),(58,60),(62,64)]])
Matrix representation of C42.291D4 ►in GL8(𝔽17)
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 7 |
| 0 | 0 | 0 | 0 | 1 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 | 0 | 16 |
| 0 | 0 | 0 | 0 | 7 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 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 |
| 0 | 0 | 16 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(8,GF(17))| [0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,7,0,0,0,0,1,0,7,0,0,0,0,0,0,7,0,16,0,0,0,0,7,0,16,0],[0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,14,3,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16] >;
C42.291D4 in GAP, Magma, Sage, TeX
C_4^2._{291}D_4 % in TeX
G:=Group("C4^2.291D4"); // GroupNames label
G:=SmallGroup(128,1971);
// by ID
G=gap.SmallGroup(128,1971);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,436,675,4037,1027,124]);
// 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*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=a^2*c^3>;
// generators/relations
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