p-group, metabelian, nilpotent (class 3), monomial, rational
Aliases: C42.290D4, C42.420C23, C4.622- 1+4, C8⋊Q8⋊16C2, D4⋊2Q8⋊10C2, C4.Q16⋊27C2, C4⋊C8.72C22, (C2×C8).72C23, C4⋊C4.177C23, (C2×C4).436C24, (C22×C4).518D4, C23.704(C2×D4), C4⋊Q8.319C22, C4.Q8.40C22, C8⋊C4.29C22, C42.6C4⋊18C2, (C4×D4).118C22, (C2×D4).180C23, C22⋊C8.63C22, (C4×Q8).115C22, (C2×Q8).168C23, C22.D8.3C2, C2.D8.106C22, D4⋊C4.50C22, C23.47D4⋊12C2, C4⋊D4.203C22, C4.121(C8.C22), C22.33(C8⋊C22), (C2×C42).897C22, Q8⋊C4.50C22, C22.696(C22×D4), C22⋊Q8.208C22, C42.28C22⋊7C2, (C22×C4).1101C23, C4.4D4.160C22, C42.C2.137C22, C23.36C23.27C2, C2.84(C23.38C23), (C2×C4⋊Q8)⋊44C2, (C2×C4).560(C2×D4), C2.64(C2×C8⋊C22), C2.64(C2×C8.C22), (C2×C4⋊C4).652C22, SmallGroup(128,1970)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.290D4
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=a2b2c3 >
Subgroups: 340 in 181 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, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C2×C4⋊C4, 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, C22×Q8, C42.6C4, D4⋊2Q8, C4.Q16, C22.D8, C23.47D4, C42.28C22, C8⋊Q8, C23.36C23, C2×C4⋊Q8, C42.290D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8⋊C22, C8.C22, C22×D4, 2- 1+4, C23.38C23, C2×C8⋊C22, C2×C8.C22, C42.290D4
Character table of C42.290D4
| 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 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ22 | 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 | orthogonal lifted from C8⋊C22 |
| ρ23 | 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 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ24 | 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 |
| ρ25 | 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 | symplectic lifted from 2- 1+4, 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 52 29 59)(2 49 30 64)(3 54 31 61)(4 51 32 58)(5 56 25 63)(6 53 26 60)(7 50 27 57)(8 55 28 62)(9 41 40 22)(10 46 33 19)(11 43 34 24)(12 48 35 21)(13 45 36 18)(14 42 37 23)(15 47 38 20)(16 44 39 17)
(1 9 5 13)(2 33 6 37)(3 11 7 15)(4 35 8 39)(10 26 14 30)(12 28 16 32)(17 51 21 55)(18 59 22 63)(19 53 23 49)(20 61 24 57)(25 36 29 40)(27 38 31 34)(41 56 45 52)(42 64 46 60)(43 50 47 54)(44 58 48 62)
(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 28)(3 7)(4 26)(6 32)(8 30)(9 36)(10 12)(11 34)(13 40)(14 16)(15 38)(17 46)(19 44)(20 24)(21 42)(23 48)(27 31)(33 35)(37 39)(43 47)(49 51)(50 57)(52 63)(53 55)(54 61)(56 59)(58 64)(60 62)
G:=sub<Sym(64)| (1,52,29,59)(2,49,30,64)(3,54,31,61)(4,51,32,58)(5,56,25,63)(6,53,26,60)(7,50,27,57)(8,55,28,62)(9,41,40,22)(10,46,33,19)(11,43,34,24)(12,48,35,21)(13,45,36,18)(14,42,37,23)(15,47,38,20)(16,44,39,17), (1,9,5,13)(2,33,6,37)(3,11,7,15)(4,35,8,39)(10,26,14,30)(12,28,16,32)(17,51,21,55)(18,59,22,63)(19,53,23,49)(20,61,24,57)(25,36,29,40)(27,38,31,34)(41,56,45,52)(42,64,46,60)(43,50,47,54)(44,58,48,62), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,36)(10,12)(11,34)(13,40)(14,16)(15,38)(17,46)(19,44)(20,24)(21,42)(23,48)(27,31)(33,35)(37,39)(43,47)(49,51)(50,57)(52,63)(53,55)(54,61)(56,59)(58,64)(60,62)>;
G:=Group( (1,52,29,59)(2,49,30,64)(3,54,31,61)(4,51,32,58)(5,56,25,63)(6,53,26,60)(7,50,27,57)(8,55,28,62)(9,41,40,22)(10,46,33,19)(11,43,34,24)(12,48,35,21)(13,45,36,18)(14,42,37,23)(15,47,38,20)(16,44,39,17), (1,9,5,13)(2,33,6,37)(3,11,7,15)(4,35,8,39)(10,26,14,30)(12,28,16,32)(17,51,21,55)(18,59,22,63)(19,53,23,49)(20,61,24,57)(25,36,29,40)(27,38,31,34)(41,56,45,52)(42,64,46,60)(43,50,47,54)(44,58,48,62), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,36)(10,12)(11,34)(13,40)(14,16)(15,38)(17,46)(19,44)(20,24)(21,42)(23,48)(27,31)(33,35)(37,39)(43,47)(49,51)(50,57)(52,63)(53,55)(54,61)(56,59)(58,64)(60,62) );
G=PermutationGroup([[(1,52,29,59),(2,49,30,64),(3,54,31,61),(4,51,32,58),(5,56,25,63),(6,53,26,60),(7,50,27,57),(8,55,28,62),(9,41,40,22),(10,46,33,19),(11,43,34,24),(12,48,35,21),(13,45,36,18),(14,42,37,23),(15,47,38,20),(16,44,39,17)], [(1,9,5,13),(2,33,6,37),(3,11,7,15),(4,35,8,39),(10,26,14,30),(12,28,16,32),(17,51,21,55),(18,59,22,63),(19,53,23,49),(20,61,24,57),(25,36,29,40),(27,38,31,34),(41,56,45,52),(42,64,46,60),(43,50,47,54),(44,58,48,62)], [(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)], [(2,28),(3,7),(4,26),(6,32),(8,30),(9,36),(10,12),(11,34),(13,40),(14,16),(15,38),(17,46),(19,44),(20,24),(21,42),(23,48),(27,31),(33,35),(37,39),(43,47),(49,51),(50,57),(52,63),(53,55),(54,61),(56,59),(58,64),(60,62)]])
Matrix representation of C42.290D4 ►in GL8(𝔽17)
| 4 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 8 | 13 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 1 | 16 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 16 | 16 | 0 |
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 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 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 1 | 16 | 0 | 0 | 0 | 0 |
| 2 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 1 | 16 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(17))| [4,0,8,0,0,0,0,0,8,13,8,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,8,4,0,0,0,0,0,0,0,0,0,16,1,16,0,0,0,0,1,0,16,16,0,0,0,0,1,16,0,16,0,0,0,0,16,16,1,0],[16,0,0,0,0,0,0,0,15,1,15,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[1,16,2,1,0,0,0,0,0,0,0,1,0,0,0,0,16,1,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,16,0,16,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C42.290D4 in GAP, Magma, Sage, TeX
C_4^2._{290}D_4 % in TeX
G:=Group("C4^2.290D4"); // GroupNames label
G:=SmallGroup(128,1970);
// by ID
G=gap.SmallGroup(128,1970);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,100,675,4037,1027,124]);
// 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=a^2*b^2*c^3>;
// generators/relations
Export