p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C42.301D4, C4.92- 1+4, C42.435C23, C4.282+ 1+4, C4⋊D8⋊30C2, C8⋊2D4⋊16C2, D4⋊2Q8⋊14C2, C4⋊C8.81C22, (C2×C8).77C23, C4⋊C4.192C23, (C2×C4).451C24, (C2×D8).75C22, (C22×C4).528D4, C23.308(C2×D4), C4⋊Q8.329C22, C4.106(C8⋊C22), C4⋊M4(2)⋊12C2, C4.Q8.47C22, (C2×D4).193C23, (C4×D4).131C22, D4⋊C4.58C22, C4⋊1D4.178C22, C4⋊D4.213C22, (C2×C42).908C22, C22.711(C22×D4), C22.26C24⋊25C2, (C22×C4).1106C23, (C2×M4(2)).89C22, C2.70(C22.31C24), (C2×C4).575(C2×D4), C2.68(C2×C8⋊C22), SmallGroup(128,1985)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.301D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1, dad=ab2, cbc-1=dbd=b-1, dcd=a2c3 >
Subgroups: 484 in 215 conjugacy classes, 88 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), D8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, D4⋊C4, C4⋊C8, C4.Q8, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×D8, C2×C4○D4, C4⋊M4(2), C4⋊D8, C8⋊2D4, D4⋊2Q8, C22.26C24, C42.301D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8⋊C22, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C2×C8⋊C22, C42.301D4
Character table of C42.301D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 0 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 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 | 0 | 0 | 4 | 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 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
►On 64 points
Generators in S64
(1 50 5 54)(2 55 6 51)(3 52 7 56)(4 49 8 53)(9 34 13 38)(10 39 14 35)(11 36 15 40)(12 33 16 37)(17 57 21 61)(18 62 22 58)(19 59 23 63)(20 64 24 60)(25 45 29 41)(26 42 30 46)(27 47 31 43)(28 44 32 48)
(1 21 42 12)(2 13 43 22)(3 23 44 14)(4 15 45 24)(5 17 46 16)(6 9 47 18)(7 19 48 10)(8 11 41 20)(25 64 53 36)(26 37 54 57)(27 58 55 38)(28 39 56 59)(29 60 49 40)(30 33 50 61)(31 62 51 34)(32 35 52 63)
(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 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 48)(24 47)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 40)(32 39)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 64)(56 63)
G:=sub<Sym(64)| (1,50,5,54)(2,55,6,51)(3,52,7,56)(4,49,8,53)(9,34,13,38)(10,39,14,35)(11,36,15,40)(12,33,16,37)(17,57,21,61)(18,62,22,58)(19,59,23,63)(20,64,24,60)(25,45,29,41)(26,42,30,46)(27,47,31,43)(28,44,32,48), (1,21,42,12)(2,13,43,22)(3,23,44,14)(4,15,45,24)(5,17,46,16)(6,9,47,18)(7,19,48,10)(8,11,41,20)(25,64,53,36)(26,37,54,57)(27,58,55,38)(28,39,56,59)(29,60,49,40)(30,33,50,61)(31,62,51,34)(32,35,52,63), (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), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,48)(24,47)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,40)(32,39)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,64)(56,63)>;
G:=Group( (1,50,5,54)(2,55,6,51)(3,52,7,56)(4,49,8,53)(9,34,13,38)(10,39,14,35)(11,36,15,40)(12,33,16,37)(17,57,21,61)(18,62,22,58)(19,59,23,63)(20,64,24,60)(25,45,29,41)(26,42,30,46)(27,47,31,43)(28,44,32,48), (1,21,42,12)(2,13,43,22)(3,23,44,14)(4,15,45,24)(5,17,46,16)(6,9,47,18)(7,19,48,10)(8,11,41,20)(25,64,53,36)(26,37,54,57)(27,58,55,38)(28,39,56,59)(29,60,49,40)(30,33,50,61)(31,62,51,34)(32,35,52,63), (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), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,48)(24,47)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,40)(32,39)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,64)(56,63) );
G=PermutationGroup([[(1,50,5,54),(2,55,6,51),(3,52,7,56),(4,49,8,53),(9,34,13,38),(10,39,14,35),(11,36,15,40),(12,33,16,37),(17,57,21,61),(18,62,22,58),(19,59,23,63),(20,64,24,60),(25,45,29,41),(26,42,30,46),(27,47,31,43),(28,44,32,48)], [(1,21,42,12),(2,13,43,22),(3,23,44,14),(4,15,45,24),(5,17,46,16),(6,9,47,18),(7,19,48,10),(8,11,41,20),(25,64,53,36),(26,37,54,57),(27,58,55,38),(28,39,56,59),(29,60,49,40),(30,33,50,61),(31,62,51,34),(32,35,52,63)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,16),(6,15),(7,14),(8,13),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,48),(24,47),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(31,40),(32,39),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,64),(56,63)]])
Matrix representation of C42.301D4 ►in GL8(𝔽17)
| 16 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 16 | 15 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 16 |
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 1 | 15 | 0 | 0 | 0 | 0 |
| 16 | 15 | 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 16 | 15 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 13 | 0 | 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 9 | 7 | 7 |
| 0 | 0 | 0 | 0 | 8 | 16 | 0 | 10 |
| 0 | 0 | 0 | 0 | 8 | 1 | 0 | 8 |
| 0 | 0 | 0 | 0 | 9 | 8 | 8 | 9 |
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 15 | 15 | 16 | 2 | 0 | 0 | 0 | 0 |
| 15 | 15 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 | 1 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 1 | 1 |
G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,16,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,16,1,0,0,0,0,0,1,0,1,0,0,0,0,1,16,0,0,0,0,0,0,0,15,0,16],[16,1,0,16,0,0,0,0,15,1,15,15,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,0,1,16,1,0,0,0,0,0,1,0,1,0,0,0,0,1,16,0,0,0,0,0,0,0,15,0,16],[13,4,0,13,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,13,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,9,8,8,9,0,0,0,0,9,16,1,8,0,0,0,0,7,0,0,8,0,0,0,0,7,10,8,9],[16,0,15,15,0,0,0,0,15,1,15,15,0,0,0,0,0,0,16,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,16,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,2,0,1] >;
C42.301D4 in GAP, Magma, Sage, TeX
C_4^2._{301}D_4 % in TeX
G:=Group("C4^2.301D4"); // GroupNames label
G:=SmallGroup(128,1985);
// by ID
G=gap.SmallGroup(128,1985);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,891,675,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=a^2*c^3>;
// generators/relations
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