p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.299D4, C4.72- 1+4, C42.433C23, C4.262+ 1+4, C8⋊D4⋊20C2, C8⋊2D4⋊15C2, C4⋊D8⋊29C2, D4.Q8⋊28C2, C4⋊SD16⋊13C2, Q8⋊Q8⋊13C2, D4⋊Q8⋊30C2, C4⋊C8.79C22, (C2×C8).75C23, D4.2D4⋊29C2, C4⋊C4.190C23, (C2×C4).449C24, (C2×D8).74C22, (C22×C4).526D4, C23.306(C2×D4), C4⋊Q8.327C22, C4.128(C8⋊C22), C4⋊M4(2)⋊10C2, C4.Q8.45C22, (C2×D4).191C23, (C4×D4).129C22, (C2×Q8).179C23, (C4×Q8).126C22, C2.D8.111C22, D4⋊C4.56C22, C4⋊D4.211C22, C4⋊1D4.177C22, (C2×C42).906C22, Q8⋊C4.54C22, (C2×SD16).40C22, C22.709(C22×D4), C22⋊Q8.216C22, C2.72(D8⋊C22), (C22×C4).1104C23, C22.26C24⋊24C2, C4.4D4.166C22, (C2×M4(2)).87C22, C42.C2.143C22, C23.36C23⋊15C2, C2.68(C22.31C24), (C2×C4).573(C2×D4), C2.67(C2×C8⋊C22), SmallGroup(128,1983)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.299D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1, dad=ab2, cbc-1=a2b-1, dbd=b-1, dcd=a2c3 >
Subgroups: 412 in 196 conjugacy classes, 86 normal (44 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×D8, C2×SD16, C2×C4○D4, C4⋊M4(2), C4⋊D8, C4⋊SD16, D4.2D4, C8⋊D4, C8⋊2D4, D4⋊Q8, Q8⋊Q8, D4.Q8, C23.36C23, C22.26C24, C42.299D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8⋊C22, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C2×C8⋊C22, D8⋊C22, C42.299D4
Character table of C42.299D4
| 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 | -4 | 0 | 0 | 4 | 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 | 0 | 0 | 4 | 0 | 0 | -4 | 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 | 0 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| 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 | -4i | 0 | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►On 64 points
(1 57 5 61)(2 62 6 58)(3 59 7 63)(4 64 8 60)(9 36 13 40)(10 33 14 37)(11 38 15 34)(12 35 16 39)(17 32 21 28)(18 29 22 25)(19 26 23 30)(20 31 24 27)(41 56 45 52)(42 53 46 49)(43 50 47 54)(44 55 48 51)
(1 25 51 16)(2 13 52 30)(3 27 53 10)(4 15 54 32)(5 29 55 12)(6 9 56 26)(7 31 49 14)(8 11 50 28)(17 60 38 47)(18 44 39 57)(19 62 40 41)(20 46 33 59)(21 64 34 43)(22 48 35 61)(23 58 36 45)(24 42 37 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 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)| (1,57,5,61)(2,62,6,58)(3,59,7,63)(4,64,8,60)(9,36,13,40)(10,33,14,37)(11,38,15,34)(12,35,16,39)(17,32,21,28)(18,29,22,25)(19,26,23,30)(20,31,24,27)(41,56,45,52)(42,53,46,49)(43,50,47,54)(44,55,48,51), (1,25,51,16)(2,13,52,30)(3,27,53,10)(4,15,54,32)(5,29,55,12)(6,9,56,26)(7,31,49,14)(8,11,50,28)(17,60,38,47)(18,44,39,57)(19,62,40,41)(20,46,33,59)(21,64,34,43)(22,48,35,61)(23,58,36,45)(24,42,37,63), (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), (1,8)(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( (1,57,5,61)(2,62,6,58)(3,59,7,63)(4,64,8,60)(9,36,13,40)(10,33,14,37)(11,38,15,34)(12,35,16,39)(17,32,21,28)(18,29,22,25)(19,26,23,30)(20,31,24,27)(41,56,45,52)(42,53,46,49)(43,50,47,54)(44,55,48,51), (1,25,51,16)(2,13,52,30)(3,27,53,10)(4,15,54,32)(5,29,55,12)(6,9,56,26)(7,31,49,14)(8,11,50,28)(17,60,38,47)(18,44,39,57)(19,62,40,41)(20,46,33,59)(21,64,34,43)(22,48,35,61)(23,58,36,45)(24,42,37,63), (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), (1,8)(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([[(1,57,5,61),(2,62,6,58),(3,59,7,63),(4,64,8,60),(9,36,13,40),(10,33,14,37),(11,38,15,34),(12,35,16,39),(17,32,21,28),(18,29,22,25),(19,26,23,30),(20,31,24,27),(41,56,45,52),(42,53,46,49),(43,50,47,54),(44,55,48,51)], [(1,25,51,16),(2,13,52,30),(3,27,53,10),(4,15,54,32),(5,29,55,12),(6,9,56,26),(7,31,49,14),(8,11,50,28),(17,60,38,47),(18,44,39,57),(19,62,40,41),(20,46,33,59),(21,64,34,43),(22,48,35,61),(23,58,36,45),(24,42,37,63)], [(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)], [(1,8),(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 C42.299D4 ►in GL8(𝔽17)
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 1 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 1 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 1 | 16 | 0 | 0 | 0 | 0 |
| 15 | 0 | 2 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 0 |
| 10 | 10 | 6 | 10 | 0 | 0 | 0 | 0 |
| 1 | 1 | 9 | 1 | 0 | 0 | 0 | 0 |
| 16 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 15 | 0 | 2 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 10 | 10 | 6 | 10 | 0 | 0 | 0 | 0 |
| 1 | 1 | 9 | 1 | 0 | 0 | 0 | 0 |
| 1 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 12 | 2 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 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 |
G:=sub<GL(8,GF(17))| [0,16,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,16,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[0,1,16,15,0,0,0,0,16,0,16,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0],[10,1,16,15,0,0,0,0,10,1,10,0,0,0,0,0,6,9,0,2,0,0,0,0,10,1,0,6,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0],[10,1,1,14,0,0,0,0,10,1,7,12,0,0,0,0,6,9,0,2,0,0,0,0,10,1,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C42.299D4 in GAP, Magma, Sage, TeX
C_4^2._{299}D_4 % in TeX
G:=Group("C4^2.299D4"); // GroupNames label
G:=SmallGroup(128,1983);
// by ID
G=gap.SmallGroup(128,1983);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,891,675,80,4037,1027,124]);
// 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=a^2*b^-1,d*b*d=b^-1,d*c*d=a^2*c^3>;
// generators/relations
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