p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.277D4, C42.405C23, C4.1112+ 1+4, C8⋊3D4⋊12C2, D4⋊D4⋊27C2, C8.2D4⋊12C2, C4⋊C8.64C22, (C2×C8).67C23, D4.2D4⋊24C2, D4.7D4⋊28C2, C4⋊C4.158C23, (C2×C4).417C24, Q8.D4⋊24C2, (C2×D8).72C22, (C22×C4).506D4, C23.289(C2×D4), C4⋊Q8.307C22, C42.6C4⋊15C2, C8⋊C4.21C22, (C4×D4).108C22, (C2×D4).166C23, C22⋊C8.52C22, (C2×Q16).72C22, (C2×Q8).154C23, (C4×Q8).105C22, D4⋊C4.46C22, C4⋊D4.195C22, C4⋊1D4.168C22, (C2×C42).884C22, Q8⋊C4.47C22, (C2×SD16).36C22, C22.677(C22×D4), C22⋊Q8.200C22, C2.60(D8⋊C22), C22.26C24⋊21C2, C42.29C22⋊2C2, C42.30C22⋊2C2, (C22×C4).1088C23, C4.4D4.157C22, C42.C2.128C22, C23.36C23⋊12C2, C2.88(C22.29C24), (C2×C4).546(C2×D4), (C2×C4○D4).176C22, SmallGroup(128,1951)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.277D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=dbd=a2b-1, dcd=b2c3 >
Subgroups: 420 in 198 conjugacy classes, 84 normal (42 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, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C2×D8, C2×SD16, C2×Q16, C2×C4○D4, C42.6C4, D4⋊D4, D4.7D4, D4.2D4, Q8.D4, C42.29C22, C42.30C22, C8⋊3D4, C8.2D4, C23.36C23, C22.26C24, C42.277D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C22.29C24, D8⋊C22, C42.277D4
Character table of C42.277D4
| 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 | 4 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 0 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -4i | 0 | 0 | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 4i | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | -4i | 0 | 0 | 4i | 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 | 0 | 4i | 0 | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►On 64 points
(1 46 51 59)(2 43 52 64)(3 48 53 61)(4 45 54 58)(5 42 55 63)(6 47 56 60)(7 44 49 57)(8 41 50 62)(9 17 26 38)(10 22 27 35)(11 19 28 40)(12 24 29 37)(13 21 30 34)(14 18 31 39)(15 23 32 36)(16 20 25 33)
(1 10 5 14)(2 32 6 28)(3 12 7 16)(4 26 8 30)(9 50 13 54)(11 52 15 56)(17 62 21 58)(18 46 22 42)(19 64 23 60)(20 48 24 44)(25 53 29 49)(27 55 31 51)(33 61 37 57)(34 45 38 41)(35 63 39 59)(36 47 40 43)
(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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 40)(25 28)(26 27)(29 32)(30 31)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 64)(49 52)(50 51)(53 56)(54 55)
G:=sub<Sym(64)| (1,46,51,59)(2,43,52,64)(3,48,53,61)(4,45,54,58)(5,42,55,63)(6,47,56,60)(7,44,49,57)(8,41,50,62)(9,17,26,38)(10,22,27,35)(11,19,28,40)(12,24,29,37)(13,21,30,34)(14,18,31,39)(15,23,32,36)(16,20,25,33), (1,10,5,14)(2,32,6,28)(3,12,7,16)(4,26,8,30)(9,50,13,54)(11,52,15,56)(17,62,21,58)(18,46,22,42)(19,64,23,60)(20,48,24,44)(25,53,29,49)(27,55,31,51)(33,61,37,57)(34,45,38,41)(35,63,39,59)(36,47,40,43), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,40)(25,28)(26,27)(29,32)(30,31)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,64)(49,52)(50,51)(53,56)(54,55)>;
G:=Group( (1,46,51,59)(2,43,52,64)(3,48,53,61)(4,45,54,58)(5,42,55,63)(6,47,56,60)(7,44,49,57)(8,41,50,62)(9,17,26,38)(10,22,27,35)(11,19,28,40)(12,24,29,37)(13,21,30,34)(14,18,31,39)(15,23,32,36)(16,20,25,33), (1,10,5,14)(2,32,6,28)(3,12,7,16)(4,26,8,30)(9,50,13,54)(11,52,15,56)(17,62,21,58)(18,46,22,42)(19,64,23,60)(20,48,24,44)(25,53,29,49)(27,55,31,51)(33,61,37,57)(34,45,38,41)(35,63,39,59)(36,47,40,43), (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), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,40)(25,28)(26,27)(29,32)(30,31)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,64)(49,52)(50,51)(53,56)(54,55) );
G=PermutationGroup([[(1,46,51,59),(2,43,52,64),(3,48,53,61),(4,45,54,58),(5,42,55,63),(6,47,56,60),(7,44,49,57),(8,41,50,62),(9,17,26,38),(10,22,27,35),(11,19,28,40),(12,24,29,37),(13,21,30,34),(14,18,31,39),(15,23,32,36),(16,20,25,33)], [(1,10,5,14),(2,32,6,28),(3,12,7,16),(4,26,8,30),(9,50,13,54),(11,52,15,56),(17,62,21,58),(18,46,22,42),(19,64,23,60),(20,48,24,44),(25,53,29,49),(27,55,31,51),(33,61,37,57),(34,45,38,41),(35,63,39,59),(36,47,40,43)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,40),(25,28),(26,27),(29,32),(30,31),(41,63),(42,62),(43,61),(44,60),(45,59),(46,58),(47,57),(48,64),(49,52),(50,51),(53,56),(54,55)]])
Matrix representation of C42.277D4 ►in GL8(𝔽17)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 3 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 14 | 0 | 0 | 0 | 0 |
| 14 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 | 2 | 2 |
| 0 | 0 | 0 | 0 | 5 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 10 | 10 |
| 0 | 0 | 0 | 0 | 16 | 0 | 12 | 0 |
| 0 | 0 | 3 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 14 | 0 | 0 | 0 | 0 |
| 3 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 | 2 | 2 |
| 0 | 0 | 0 | 0 | 5 | 10 | 16 | 15 |
| 0 | 0 | 0 | 0 | 2 | 2 | 10 | 10 |
| 0 | 0 | 0 | 0 | 16 | 15 | 12 | 7 |
G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,14,5,0,0,0,0,0,0,12,3,0,0,0,0,3,12,0,0,0,0,0,0,5,14,0,0,0,0,0,0,0,0,0,0,7,5,2,16,0,0,0,0,7,0,2,0,0,0,0,0,2,16,10,12,0,0,0,0,2,0,10,0],[0,0,3,12,0,0,0,0,0,0,5,14,0,0,0,0,3,12,0,0,0,0,0,0,5,14,0,0,0,0,0,0,0,0,0,0,7,5,2,16,0,0,0,0,7,10,2,15,0,0,0,0,2,16,10,12,0,0,0,0,2,15,10,7] >;
C42.277D4 in GAP, Magma, Sage, TeX
C_4^2._{277}D_4 % in TeX
G:=Group("C4^2.277D4"); // GroupNames label
G:=SmallGroup(128,1951);
// by ID
G=gap.SmallGroup(128,1951);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,675,248,4037,1027,124]);
// 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*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=b^2*c^3>;
// generators/relations
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