p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.300D4, C4.82- 1+4, C42.434C23, C4.272+ 1+4, C8⋊D4.2C2, Q8.Q8⋊28C2, C4.Q16⋊30C2, D4⋊2Q8⋊13C2, C4⋊2Q16⋊30C2, C8.D4⋊15C2, C4⋊C8.80C22, (C2×C8).76C23, D4.D4⋊13C2, C4⋊C4.191C23, (C2×C4).450C24, Q8.D4⋊29C2, C23.307(C2×D4), (C22×C4).527D4, C4⋊Q8.328C22, C4⋊M4(2)⋊11C2, C4.Q8.46C22, (C2×D4).192C23, (C4×D4).130C22, (C2×Q8).180C23, (C2×Q16).74C22, (C4×Q8).127C22, C2.D8.112C22, D4⋊C4.57C22, C4⋊D4.212C22, C4.122(C8.C22), (C2×C42).907C22, Q8⋊C4.55C22, (C2×SD16).41C22, C22.710(C22×D4), C22⋊Q8.217C22, C2.73(D8⋊C22), (C22×C4).1105C23, C4.4D4.167C22, (C2×M4(2)).88C22, C42.C2.144C22, C23.37C23⋊26C2, C23.36C23.31C2, C2.69(C22.31C24), (C2×C4).574(C2×D4), C2.67(C2×C8.C22), SmallGroup(128,1984)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.300D4
G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c3 >
Subgroups: 316 in 174 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), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C2×M4(2), C2×SD16, C2×Q16, C4⋊M4(2), D4.D4, C4⋊2Q16, Q8.D4, C8⋊D4, C8.D4, D4⋊2Q8, C4.Q16, Q8.Q8, C23.36C23, C23.37C23, C42.300D4
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, C42.300D4
Character table of C42.300D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 0 | 2 | -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 | 0 | -2 | -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 | 0 | -2 | 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 | 0 | 2 | 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 | -4 | 0 | 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 | 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 |
| 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 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 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 | 4i | 0 | 0 | -4i | 0 | 0 | 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 | -4i | 0 | 0 | 4i | 0 | 0 | 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 38 13 34)(10 35 14 39)(11 40 15 36)(12 37 16 33)(17 28 21 32)(18 25 22 29)(19 30 23 26)(20 27 24 31)(41 52 45 56)(42 49 46 53)(43 54 47 50)(44 51 48 55)
(1 25 55 13)(2 10 56 30)(3 27 49 15)(4 12 50 32)(5 29 51 9)(6 14 52 26)(7 31 53 11)(8 16 54 28)(17 64 37 43)(18 48 38 61)(19 58 39 45)(20 42 40 63)(21 60 33 47)(22 44 34 57)(23 62 35 41)(24 46 36 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 4 5 8)(2 7 6 3)(9 16 13 12)(10 11 14 15)(17 38 21 34)(18 33 22 37)(19 36 23 40)(20 39 24 35)(25 32 29 28)(26 27 30 31)(41 63 45 59)(42 58 46 62)(43 61 47 57)(44 64 48 60)(49 56 53 52)(50 51 54 55)
G:=sub<Sym(64)| (1,57,5,61)(2,62,6,58)(3,59,7,63)(4,64,8,60)(9,38,13,34)(10,35,14,39)(11,40,15,36)(12,37,16,33)(17,28,21,32)(18,25,22,29)(19,30,23,26)(20,27,24,31)(41,52,45,56)(42,49,46,53)(43,54,47,50)(44,51,48,55), (1,25,55,13)(2,10,56,30)(3,27,49,15)(4,12,50,32)(5,29,51,9)(6,14,52,26)(7,31,53,11)(8,16,54,28)(17,64,37,43)(18,48,38,61)(19,58,39,45)(20,42,40,63)(21,60,33,47)(22,44,34,57)(23,62,35,41)(24,46,36,59), (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), (1,4,5,8)(2,7,6,3)(9,16,13,12)(10,11,14,15)(17,38,21,34)(18,33,22,37)(19,36,23,40)(20,39,24,35)(25,32,29,28)(26,27,30,31)(41,63,45,59)(42,58,46,62)(43,61,47,57)(44,64,48,60)(49,56,53,52)(50,51,54,55)>;
G:=Group( (1,57,5,61)(2,62,6,58)(3,59,7,63)(4,64,8,60)(9,38,13,34)(10,35,14,39)(11,40,15,36)(12,37,16,33)(17,28,21,32)(18,25,22,29)(19,30,23,26)(20,27,24,31)(41,52,45,56)(42,49,46,53)(43,54,47,50)(44,51,48,55), (1,25,55,13)(2,10,56,30)(3,27,49,15)(4,12,50,32)(5,29,51,9)(6,14,52,26)(7,31,53,11)(8,16,54,28)(17,64,37,43)(18,48,38,61)(19,58,39,45)(20,42,40,63)(21,60,33,47)(22,44,34,57)(23,62,35,41)(24,46,36,59), (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), (1,4,5,8)(2,7,6,3)(9,16,13,12)(10,11,14,15)(17,38,21,34)(18,33,22,37)(19,36,23,40)(20,39,24,35)(25,32,29,28)(26,27,30,31)(41,63,45,59)(42,58,46,62)(43,61,47,57)(44,64,48,60)(49,56,53,52)(50,51,54,55) );
G=PermutationGroup([[(1,57,5,61),(2,62,6,58),(3,59,7,63),(4,64,8,60),(9,38,13,34),(10,35,14,39),(11,40,15,36),(12,37,16,33),(17,28,21,32),(18,25,22,29),(19,30,23,26),(20,27,24,31),(41,52,45,56),(42,49,46,53),(43,54,47,50),(44,51,48,55)], [(1,25,55,13),(2,10,56,30),(3,27,49,15),(4,12,50,32),(5,29,51,9),(6,14,52,26),(7,31,53,11),(8,16,54,28),(17,64,37,43),(18,48,38,61),(19,58,39,45),(20,42,40,63),(21,60,33,47),(22,44,34,57),(23,62,35,41),(24,46,36,59)], [(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)], [(1,4,5,8),(2,7,6,3),(9,16,13,12),(10,11,14,15),(17,38,21,34),(18,33,22,37),(19,36,23,40),(20,39,24,35),(25,32,29,28),(26,27,30,31),(41,63,45,59),(42,58,46,62),(43,61,47,57),(44,64,48,60),(49,56,53,52),(50,51,54,55)]])
Matrix representation of C42.300D4 ►in GL8(𝔽17)
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 13 | 4 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 4 | 4 |
| 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 16 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 16 | 16 |
| 0 | 0 | 16 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 | 0 | 0 |
| 16 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 13 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 4 | 13 | 13 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 16 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 | 0 | 0 |
| 1 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 13 | 13 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 0 | 4 | 13 | 13 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 0 |
G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,13,0,4,13,0,0,0,0,4,4,0,4,0,0,0,0,8,0,0,4],[0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,16,1,0,0,0,0,16,16,0,16,0,0,0,0,15,0,0,16],[0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,13,0,13,0,0,0,0,0,13,4,13,0],[0,0,1,10,0,0,0,0,0,0,10,16,0,0,0,0,16,7,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,13,0,0,0,0,13,0,13,0,0,0,0,0,13,13,13,0] >;
C42.300D4 in GAP, Magma, Sage, TeX
C_4^2._{300}D_4 % in TeX
G:=Group("C4^2.300D4"); // GroupNames label
G:=SmallGroup(128,1984);
// by ID
G=gap.SmallGroup(128,1984);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,891,675,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
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