p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C42.276D4, C42.404C23, C4.1102+ 1+4, C4⋊2Q16⋊27C2, C8.2D4⋊11C2, C4⋊C8.63C22, (C2×C8).66C23, Q8⋊D4.3C2, D4.D4⋊10C2, C4⋊C4.157C23, (C2×C4).416C24, C22⋊Q16⋊22C2, C23.696(C2×D4), (C22×C4).505D4, C4⋊Q8.306C22, C42.6C4⋊14C2, C8⋊C4.20C22, (C2×D4).165C23, (C4×D4).107C22, C22⋊C8.51C22, (C2×Q8).153C23, (C2×Q16).71C22, (C4×Q8).104C22, C4⋊D4.194C22, C4.120(C8.C22), (C2×C42).883C22, Q8⋊C4.46C22, (C2×SD16).35C22, C22.676(C22×D4), C22⋊Q8.199C22, C42.30C22⋊1C2, (C22×C4).1087C23, C4.4D4.156C22, C22.21(C8.C22), C42.C2.127C22, (C22×Q8).324C22, C2.87(C22.29C24), C23.36C23.24C2, (C2×C4⋊Q8)⋊41C2, (C2×C4).545(C2×D4), C2.58(C2×C8.C22), SmallGroup(128,1950)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.276D4
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=c3 >
Subgroups: 372 in 193 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, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, Q8⋊C4, C4⋊C8, C2×C42, 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, C2×SD16, C2×Q16, C22×Q8, C42.6C4, Q8⋊D4, C22⋊Q16, D4.D4, C4⋊2Q16, C42.30C22, C8.2D4, C23.36C23, C2×C4⋊Q8, C42.276D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8.C22, C22×D4, 2+ 1+4, C22.29C24, C2×C8.C22, C42.276D4
Character table of C42.276D4
| 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 | 0 | 0 | 4 | -4 | 0 | 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 | 4 | 0 | 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 | 4 | 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 |
| 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 | 4 | -4 | 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 |
►On 64 points
(1 63 29 50)(2 60 30 55)(3 57 31 52)(4 62 32 49)(5 59 25 54)(6 64 26 51)(7 61 27 56)(8 58 28 53)(9 22 35 42)(10 19 36 47)(11 24 37 44)(12 21 38 41)(13 18 39 46)(14 23 40 43)(15 20 33 48)(16 17 34 45)
(1 36 5 40)(2 11 6 15)(3 38 7 34)(4 13 8 9)(10 25 14 29)(12 27 16 31)(17 52 21 56)(18 58 22 62)(19 54 23 50)(20 60 24 64)(26 33 30 37)(28 35 32 39)(41 61 45 57)(42 49 46 53)(43 63 47 59)(44 51 48 55)
(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 4)(3 7)(6 8)(9 37)(10 40)(11 35)(12 38)(13 33)(14 36)(15 39)(16 34)(17 21)(18 24)(20 22)(26 28)(27 31)(30 32)(41 45)(42 48)(44 46)(49 64)(50 59)(51 62)(52 57)(53 60)(54 63)(55 58)(56 61)
G:=sub<Sym(64)| (1,63,29,50)(2,60,30,55)(3,57,31,52)(4,62,32,49)(5,59,25,54)(6,64,26,51)(7,61,27,56)(8,58,28,53)(9,22,35,42)(10,19,36,47)(11,24,37,44)(12,21,38,41)(13,18,39,46)(14,23,40,43)(15,20,33,48)(16,17,34,45), (1,36,5,40)(2,11,6,15)(3,38,7,34)(4,13,8,9)(10,25,14,29)(12,27,16,31)(17,52,21,56)(18,58,22,62)(19,54,23,50)(20,60,24,64)(26,33,30,37)(28,35,32,39)(41,61,45,57)(42,49,46,53)(43,63,47,59)(44,51,48,55), (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), (2,4)(3,7)(6,8)(9,37)(10,40)(11,35)(12,38)(13,33)(14,36)(15,39)(16,34)(17,21)(18,24)(20,22)(26,28)(27,31)(30,32)(41,45)(42,48)(44,46)(49,64)(50,59)(51,62)(52,57)(53,60)(54,63)(55,58)(56,61)>;
G:=Group( (1,63,29,50)(2,60,30,55)(3,57,31,52)(4,62,32,49)(5,59,25,54)(6,64,26,51)(7,61,27,56)(8,58,28,53)(9,22,35,42)(10,19,36,47)(11,24,37,44)(12,21,38,41)(13,18,39,46)(14,23,40,43)(15,20,33,48)(16,17,34,45), (1,36,5,40)(2,11,6,15)(3,38,7,34)(4,13,8,9)(10,25,14,29)(12,27,16,31)(17,52,21,56)(18,58,22,62)(19,54,23,50)(20,60,24,64)(26,33,30,37)(28,35,32,39)(41,61,45,57)(42,49,46,53)(43,63,47,59)(44,51,48,55), (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), (2,4)(3,7)(6,8)(9,37)(10,40)(11,35)(12,38)(13,33)(14,36)(15,39)(16,34)(17,21)(18,24)(20,22)(26,28)(27,31)(30,32)(41,45)(42,48)(44,46)(49,64)(50,59)(51,62)(52,57)(53,60)(54,63)(55,58)(56,61) );
G=PermutationGroup([[(1,63,29,50),(2,60,30,55),(3,57,31,52),(4,62,32,49),(5,59,25,54),(6,64,26,51),(7,61,27,56),(8,58,28,53),(9,22,35,42),(10,19,36,47),(11,24,37,44),(12,21,38,41),(13,18,39,46),(14,23,40,43),(15,20,33,48),(16,17,34,45)], [(1,36,5,40),(2,11,6,15),(3,38,7,34),(4,13,8,9),(10,25,14,29),(12,27,16,31),(17,52,21,56),(18,58,22,62),(19,54,23,50),(20,60,24,64),(26,33,30,37),(28,35,32,39),(41,61,45,57),(42,49,46,53),(43,63,47,59),(44,51,48,55)], [(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)], [(2,4),(3,7),(6,8),(9,37),(10,40),(11,35),(12,38),(13,33),(14,36),(15,39),(16,34),(17,21),(18,24),(20,22),(26,28),(27,31),(30,32),(41,45),(42,48),(44,46),(49,64),(50,59),(51,62),(52,57),(53,60),(54,63),(55,58),(56,61)]])
Matrix representation of C42.276D4 ►in GL8(𝔽17)
| 0 | 0 | 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 14 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 14 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 3 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 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 | 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 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 | 1 |
G:=sub<GL(8,GF(17))| [0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,0,0,5,3,0,0,0,0,0,5,0,0,3,0,0,0,0,14,0,0,12,0,0,0,0,0,14,12,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0],[0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1] >;
C42.276D4 in GAP, Magma, Sage, TeX
C_4^2._{276}D_4 % in TeX
G:=Group("C4^2.276D4"); // GroupNames label
G:=SmallGroup(128,1950);
// by ID
G=gap.SmallGroup(128,1950);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,352,675,4037,1027,124]);
// 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=c^3>;
// generators/relations
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