p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.274D4, C42.402C23, C4.1082+ 1+4, C4⋊SD16⋊9C2, C4⋊2Q16⋊26C2, C8.2D4⋊10C2, C4⋊C8.62C22, (C2×C8).64C23, D4.7D4⋊27C2, C4⋊C4.155C23, (C2×C4).414C24, (C22×C4).503D4, C23.288(C2×D4), C4⋊Q8.305C22, C8⋊C4.19C22, C42.6C4⋊12C2, (C2×D4).163C23, C22⋊C8.49C22, (C4×Q8).103C22, (C2×Q16).70C22, (C2×Q8).151C23, D4⋊C4.45C22, C4⋊1D4.166C22, C4.100(C8.C22), (C2×C42).881C22, Q8⋊C4.45C22, (C2×SD16).34C22, C22.674(C22×D4), C22⋊Q8.197C22, C2.59(D8⋊C22), (C22×C4).1085C23, C42.28C22⋊4C2, C4.4D4.155C22, C23.37C23⋊19C2, C2.85(C22.29C24), C22.26C24.42C2, (C2×C4).543(C2×D4), C2.57(C2×C8.C22), (C2×C4○D4).175C22, SmallGroup(128,1948)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.274D4
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=c3 >
Subgroups: 396 in 195 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, SD16, Q16, C22×C4, C22×C4, 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×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C42.C2, C4⋊1D4, C4⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C42.6C4, D4.7D4, C4⋊SD16, C4⋊2Q16, C42.28C22, C8.2D4, C22.26C24, C23.37C23, C42.274D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8.C22, C22×D4, 2+ 1+4, C22.29C24, C2×C8.C22, D8⋊C22, C42.274D4
Character table of C42.274D4
| 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 | 0 | 0 | 4 | 0 | -4 | 0 | 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 | -4 | 0 | 4 | 0 | 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 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 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 | 0 | -4i | 0 | 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 | 4i | 0 | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►On 64 points
(1 26 63 11)(2 31 64 16)(3 28 57 13)(4 25 58 10)(5 30 59 15)(6 27 60 12)(7 32 61 9)(8 29 62 14)(17 34 56 48)(18 39 49 45)(19 36 50 42)(20 33 51 47)(21 38 52 44)(22 35 53 41)(23 40 54 46)(24 37 55 43)
(1 24 5 20)(2 56 6 52)(3 18 7 22)(4 50 8 54)(9 41 13 45)(10 36 14 40)(11 43 15 47)(12 38 16 34)(17 60 21 64)(19 62 23 58)(25 42 29 46)(26 37 30 33)(27 44 31 48)(28 39 32 35)(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 40 5 36)(2 35 6 39)(3 38 7 34)(4 33 8 37)(9 56 13 52)(10 51 14 55)(11 54 15 50)(12 49 16 53)(17 28 21 32)(18 31 22 27)(19 26 23 30)(20 29 24 25)(41 60 45 64)(42 63 46 59)(43 58 47 62)(44 61 48 57)
G:=sub<Sym(64)| (1,26,63,11)(2,31,64,16)(3,28,57,13)(4,25,58,10)(5,30,59,15)(6,27,60,12)(7,32,61,9)(8,29,62,14)(17,34,56,48)(18,39,49,45)(19,36,50,42)(20,33,51,47)(21,38,52,44)(22,35,53,41)(23,40,54,46)(24,37,55,43), (1,24,5,20)(2,56,6,52)(3,18,7,22)(4,50,8,54)(9,41,13,45)(10,36,14,40)(11,43,15,47)(12,38,16,34)(17,60,21,64)(19,62,23,58)(25,42,29,46)(26,37,30,33)(27,44,31,48)(28,39,32,35)(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), (1,40,5,36)(2,35,6,39)(3,38,7,34)(4,33,8,37)(9,56,13,52)(10,51,14,55)(11,54,15,50)(12,49,16,53)(17,28,21,32)(18,31,22,27)(19,26,23,30)(20,29,24,25)(41,60,45,64)(42,63,46,59)(43,58,47,62)(44,61,48,57)>;
G:=Group( (1,26,63,11)(2,31,64,16)(3,28,57,13)(4,25,58,10)(5,30,59,15)(6,27,60,12)(7,32,61,9)(8,29,62,14)(17,34,56,48)(18,39,49,45)(19,36,50,42)(20,33,51,47)(21,38,52,44)(22,35,53,41)(23,40,54,46)(24,37,55,43), (1,24,5,20)(2,56,6,52)(3,18,7,22)(4,50,8,54)(9,41,13,45)(10,36,14,40)(11,43,15,47)(12,38,16,34)(17,60,21,64)(19,62,23,58)(25,42,29,46)(26,37,30,33)(27,44,31,48)(28,39,32,35)(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), (1,40,5,36)(2,35,6,39)(3,38,7,34)(4,33,8,37)(9,56,13,52)(10,51,14,55)(11,54,15,50)(12,49,16,53)(17,28,21,32)(18,31,22,27)(19,26,23,30)(20,29,24,25)(41,60,45,64)(42,63,46,59)(43,58,47,62)(44,61,48,57) );
G=PermutationGroup([[(1,26,63,11),(2,31,64,16),(3,28,57,13),(4,25,58,10),(5,30,59,15),(6,27,60,12),(7,32,61,9),(8,29,62,14),(17,34,56,48),(18,39,49,45),(19,36,50,42),(20,33,51,47),(21,38,52,44),(22,35,53,41),(23,40,54,46),(24,37,55,43)], [(1,24,5,20),(2,56,6,52),(3,18,7,22),(4,50,8,54),(9,41,13,45),(10,36,14,40),(11,43,15,47),(12,38,16,34),(17,60,21,64),(19,62,23,58),(25,42,29,46),(26,37,30,33),(27,44,31,48),(28,39,32,35),(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)], [(1,40,5,36),(2,35,6,39),(3,38,7,34),(4,33,8,37),(9,56,13,52),(10,51,14,55),(11,54,15,50),(12,49,16,53),(17,28,21,32),(18,31,22,27),(19,26,23,30),(20,29,24,25),(41,60,45,64),(42,63,46,59),(43,58,47,62),(44,61,48,57)]])
Matrix representation of C42.274D4 ►in GL8(𝔽17)
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 15 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 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 | 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 |
| 0 | 0 | 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 9 | 15 | 15 |
| 0 | 0 | 0 | 0 | 8 | 9 | 15 | 2 |
| 0 | 0 | 0 | 0 | 15 | 2 | 8 | 9 |
| 0 | 0 | 0 | 0 | 2 | 2 | 8 | 8 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 2 | 8 | 9 |
| 0 | 0 | 0 | 0 | 2 | 2 | 8 | 8 |
| 0 | 0 | 0 | 0 | 9 | 9 | 15 | 15 |
| 0 | 0 | 0 | 0 | 8 | 9 | 15 | 2 |
G:=sub<GL(8,GF(17))| [1,15,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,16,0,0,0,0,0,0,0,16,1,0,0,0,0,16,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,0,0,0,9,8,15,2,0,0,0,0,9,9,2,2,0,0,0,0,15,15,8,8,0,0,0,0,15,2,9,8],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,15,2,9,8,0,0,0,0,2,2,9,9,0,0,0,0,8,8,15,15,0,0,0,0,9,8,15,2] >;
C42.274D4 in GAP, Magma, Sage, TeX
C_4^2._{274}D_4 % in TeX
G:=Group("C4^2.274D4"); // GroupNames label
G:=SmallGroup(128,1948);
// by ID
G=gap.SmallGroup(128,1948);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,891,675,248,4037,1027,124]);
// 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=c^3>;
// generators/relations
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