p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.304D4, C42.438C23, C4.122- 1+4, C4.312+ 1+4, C8⋊D4⋊22C2, C8⋊2D4.6C2, Q8.Q8⋊29C2, D4.Q8⋊29C2, C8.D4⋊16C2, C4⋊C8.84C22, (C2×C8).80C23, D4.2D4⋊30C2, C4⋊C4.195C23, (C2×C4).454C24, Q8.D4⋊30C2, (C2×D8).76C22, (C22×C4).531D4, C23.311(C2×D4), C4⋊M4(2)⋊14C2, C4.Q8.49C22, (C2×D4).195C23, (C4×D4).133C22, (C4×Q8).130C22, (C2×Q8).183C23, (C2×Q16).76C22, C2.D8.114C22, D4⋊C4.60C22, C4⋊D4.215C22, (C2×C42).911C22, Q8⋊C4.58C22, (C2×SD16).43C22, C22.714(C22×D4), C22⋊Q8.220C22, C2.74(D8⋊C22), (C22×C4).1109C23, C4.4D4.168C22, (C2×M4(2)).92C22, C42.C2.145C22, C23.36C23⋊16C2, C2.73(C22.31C24), (C2×C4).578(C2×D4), SmallGroup(128,1988)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.304D4
G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=dbd=a2b-1, dcd=a2c3 >
Subgroups: 340 in 177 conjugacy classes, 84 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4⋊M4(2), D4.2D4, Q8.D4, C8⋊D4, C8⋊2D4, C8.D4, D4.Q8, Q8.Q8, C23.36C23, C42.304D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, D8⋊C22, C42.304D4
Character table of C42.304D4
| 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 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 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 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 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 | 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 | 0 | 0 | 4 | 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 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -4i | 0 | 0 | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | -4i | 0 | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 4i | 0 | 0 | -4i | 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 | 0 | 0 | 4i | 0 | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►On 64 points
(1 63 5 59)(2 60 6 64)(3 57 7 61)(4 62 8 58)(9 34 13 38)(10 39 14 35)(11 36 15 40)(12 33 16 37)(17 26 21 30)(18 31 22 27)(19 28 23 32)(20 25 24 29)(41 50 45 54)(42 55 46 51)(43 52 47 56)(44 49 48 53)
(1 31 51 14)(2 11 52 28)(3 25 53 16)(4 13 54 30)(5 27 55 10)(6 15 56 32)(7 29 49 12)(8 9 50 26)(17 62 38 41)(18 46 39 59)(19 64 40 43)(20 48 33 61)(21 58 34 45)(22 42 35 63)(23 60 36 47)(24 44 37 57)
(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)| (1,63,5,59)(2,60,6,64)(3,57,7,61)(4,62,8,58)(9,34,13,38)(10,39,14,35)(11,36,15,40)(12,33,16,37)(17,26,21,30)(18,31,22,27)(19,28,23,32)(20,25,24,29)(41,50,45,54)(42,55,46,51)(43,52,47,56)(44,49,48,53), (1,31,51,14)(2,11,52,28)(3,25,53,16)(4,13,54,30)(5,27,55,10)(6,15,56,32)(7,29,49,12)(8,9,50,26)(17,62,38,41)(18,46,39,59)(19,64,40,43)(20,48,33,61)(21,58,34,45)(22,42,35,63)(23,60,36,47)(24,44,37,57), (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( (1,63,5,59)(2,60,6,64)(3,57,7,61)(4,62,8,58)(9,34,13,38)(10,39,14,35)(11,36,15,40)(12,33,16,37)(17,26,21,30)(18,31,22,27)(19,28,23,32)(20,25,24,29)(41,50,45,54)(42,55,46,51)(43,52,47,56)(44,49,48,53), (1,31,51,14)(2,11,52,28)(3,25,53,16)(4,13,54,30)(5,27,55,10)(6,15,56,32)(7,29,49,12)(8,9,50,26)(17,62,38,41)(18,46,39,59)(19,64,40,43)(20,48,33,61)(21,58,34,45)(22,42,35,63)(23,60,36,47)(24,44,37,57), (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([[(1,63,5,59),(2,60,6,64),(3,57,7,61),(4,62,8,58),(9,34,13,38),(10,39,14,35),(11,36,15,40),(12,33,16,37),(17,26,21,30),(18,31,22,27),(19,28,23,32),(20,25,24,29),(41,50,45,54),(42,55,46,51),(43,52,47,56),(44,49,48,53)], [(1,31,51,14),(2,11,52,28),(3,25,53,16),(4,13,54,30),(5,27,55,10),(6,15,56,32),(7,29,49,12),(8,9,50,26),(17,62,38,41),(18,46,39,59),(19,64,40,43),(20,48,33,61),(21,58,34,45),(22,42,35,63),(23,60,36,47),(24,44,37,57)], [(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 C42.304D4 ►in GL8(𝔽17)
| 0 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 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 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 | 3 | 3 |
| 0 | 0 | 0 | 0 | 16 | 16 | 14 | 3 |
| 0 | 0 | 0 | 0 | 3 | 3 | 1 | 16 |
| 0 | 0 | 0 | 0 | 14 | 3 | 1 | 1 |
| 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 | 16 | 1 | 3 | 3 |
| 0 | 0 | 0 | 0 | 1 | 1 | 3 | 14 |
| 0 | 0 | 0 | 0 | 14 | 14 | 16 | 1 |
| 0 | 0 | 0 | 0 | 14 | 3 | 1 | 1 |
G:=sub<GL(8,GF(17))| [0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,0,12,12,0,0,0,0,0,0,5,12,0,0,0,0,5,5,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,0,0,16,16,3,14,0,0,0,0,1,16,3,3,0,0,0,0,3,14,1,1,0,0,0,0,3,3,16,1],[0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,16,1,14,14,0,0,0,0,1,1,14,3,0,0,0,0,3,3,16,1,0,0,0,0,3,14,1,1] >;
C42.304D4 in GAP, Magma, Sage, TeX
C_4^2._{304}D_4 % in TeX
G:=Group("C4^2.304D4"); // GroupNames label
G:=SmallGroup(128,1988);
// by ID
G=gap.SmallGroup(128,1988);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,891,675,304,4037,1027,124]);
// 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^-1*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=a^2*c^3>;
// generators/relations
Export