p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.292D4, C42.422C23, C4.642- 1+4, C8⋊Q8⋊18C2, D4.Q8⋊24C2, Q8.Q8⋊24C2, C4⋊C8.74C22, (C2×C8).74C23, C4⋊C4.179C23, (C2×C4).438C24, C23.301(C2×D4), (C22×C4).520D4, C4⋊Q8.321C22, C8⋊C4.31C22, C42.6C4⋊20C2, C4.Q8.42C22, (C4×D4).120C22, (C2×D4).182C23, C22⋊C8.65C22, (C2×Q8).170C23, (C4×Q8).117C22, C2.D8.108C22, D4⋊C4.52C22, C4⋊D4.205C22, C23.20D4⋊28C2, (C2×C42).899C22, Q8⋊C4.52C22, C23.19D4.3C2, C22.698(C22×D4), C22⋊Q8.210C22, C2.67(D8⋊C22), (C22×C4).1103C23, C42.28C22⋊9C2, C4.4D4.162C22, C42.C2.139C22, C42⋊C2.168C22, C23.37C23⋊23C2, C23.36C23.29C2, C2.86(C23.38C23), (C2×C4).562(C2×D4), SmallGroup(128,1972)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.292D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=dbd=a2b-1, dcd=a2c3 >
Subgroups: 292 in 164 conjugacy classes, 84 normal (42 characteristic)
C1, C2, C2, 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, 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, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C42.6C4, D4.Q8, Q8.Q8, C23.19D4, C23.20D4, C42.28C22, C8⋊Q8, C23.36C23, C23.37C23, C42.292D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2- 1+4, C23.38C23, D8⋊C22, C42.292D4
Character table of C42.292D4
| 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 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 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 | 4i | 0 | 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 | 4i | 0 | 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 | -4i | 0 | 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 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►On 64 points
(1 61 25 46)(2 58 26 43)(3 63 27 48)(4 60 28 45)(5 57 29 42)(6 62 30 47)(7 59 31 44)(8 64 32 41)(9 20 40 56)(10 17 33 53)(11 22 34 50)(12 19 35 55)(13 24 36 52)(14 21 37 49)(15 18 38 54)(16 23 39 51)
(1 38 5 34)(2 12 6 16)(3 40 7 36)(4 14 8 10)(9 31 13 27)(11 25 15 29)(17 60 21 64)(18 42 22 46)(19 62 23 58)(20 44 24 48)(26 35 30 39)(28 37 32 33)(41 53 45 49)(43 55 47 51)(50 61 54 57)(52 63 56 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)
(2 28)(3 7)(4 26)(6 32)(8 30)(9 40)(10 12)(11 38)(13 36)(14 16)(15 34)(17 51)(19 49)(20 24)(21 55)(23 53)(27 31)(33 35)(37 39)(41 43)(42 61)(44 59)(45 47)(46 57)(48 63)(52 56)(58 64)(60 62)
G:=sub<Sym(64)| (1,61,25,46)(2,58,26,43)(3,63,27,48)(4,60,28,45)(5,57,29,42)(6,62,30,47)(7,59,31,44)(8,64,32,41)(9,20,40,56)(10,17,33,53)(11,22,34,50)(12,19,35,55)(13,24,36,52)(14,21,37,49)(15,18,38,54)(16,23,39,51), (1,38,5,34)(2,12,6,16)(3,40,7,36)(4,14,8,10)(9,31,13,27)(11,25,15,29)(17,60,21,64)(18,42,22,46)(19,62,23,58)(20,44,24,48)(26,35,30,39)(28,37,32,33)(41,53,45,49)(43,55,47,51)(50,61,54,57)(52,63,56,59), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,40)(10,12)(11,38)(13,36)(14,16)(15,34)(17,51)(19,49)(20,24)(21,55)(23,53)(27,31)(33,35)(37,39)(41,43)(42,61)(44,59)(45,47)(46,57)(48,63)(52,56)(58,64)(60,62)>;
G:=Group( (1,61,25,46)(2,58,26,43)(3,63,27,48)(4,60,28,45)(5,57,29,42)(6,62,30,47)(7,59,31,44)(8,64,32,41)(9,20,40,56)(10,17,33,53)(11,22,34,50)(12,19,35,55)(13,24,36,52)(14,21,37,49)(15,18,38,54)(16,23,39,51), (1,38,5,34)(2,12,6,16)(3,40,7,36)(4,14,8,10)(9,31,13,27)(11,25,15,29)(17,60,21,64)(18,42,22,46)(19,62,23,58)(20,44,24,48)(26,35,30,39)(28,37,32,33)(41,53,45,49)(43,55,47,51)(50,61,54,57)(52,63,56,59), (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), (2,28)(3,7)(4,26)(6,32)(8,30)(9,40)(10,12)(11,38)(13,36)(14,16)(15,34)(17,51)(19,49)(20,24)(21,55)(23,53)(27,31)(33,35)(37,39)(41,43)(42,61)(44,59)(45,47)(46,57)(48,63)(52,56)(58,64)(60,62) );
G=PermutationGroup([[(1,61,25,46),(2,58,26,43),(3,63,27,48),(4,60,28,45),(5,57,29,42),(6,62,30,47),(7,59,31,44),(8,64,32,41),(9,20,40,56),(10,17,33,53),(11,22,34,50),(12,19,35,55),(13,24,36,52),(14,21,37,49),(15,18,38,54),(16,23,39,51)], [(1,38,5,34),(2,12,6,16),(3,40,7,36),(4,14,8,10),(9,31,13,27),(11,25,15,29),(17,60,21,64),(18,42,22,46),(19,62,23,58),(20,44,24,48),(26,35,30,39),(28,37,32,33),(41,53,45,49),(43,55,47,51),(50,61,54,57),(52,63,56,59)], [(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)], [(2,28),(3,7),(4,26),(6,32),(8,30),(9,40),(10,12),(11,38),(13,36),(14,16),(15,34),(17,51),(19,49),(20,24),(21,55),(23,53),(27,31),(33,35),(37,39),(41,43),(42,61),(44,59),(45,47),(46,57),(48,63),(52,56),(58,64),(60,62)]])
Matrix representation of C42.292D4 ►in GL8(𝔽17)
| 9 | 9 | 0 | 1 | 0 | 0 | 0 | 0 |
| 13 | 13 | 16 | 16 | 0 | 0 | 0 | 0 |
| 9 | 9 | 12 | 0 | 0 | 0 | 0 | 0 |
| 5 | 6 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 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 | 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 |
| 9 | 16 | 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 13 | 12 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 13 | 12 | 12 |
| 0 | 0 | 0 | 0 | 4 | 4 | 5 | 12 |
| 0 | 0 | 0 | 0 | 12 | 12 | 13 | 4 |
| 0 | 0 | 0 | 0 | 5 | 12 | 13 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 16 | 16 | 15 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 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 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(17))| [9,13,9,5,0,0,0,0,9,13,9,6,0,0,0,0,0,16,12,9,0,0,0,0,1,16,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[9,0,1,6,0,0,0,0,16,0,0,13,0,0,0,0,16,1,0,12,0,0,0,0,15,0,0,8,0,0,0,0,0,0,0,0,4,4,12,5,0,0,0,0,13,4,12,12,0,0,0,0,12,5,13,13,0,0,0,0,12,12,4,13],[1,0,9,0,0,0,0,0,0,16,16,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1] >;
C42.292D4 in GAP, Magma, Sage, TeX
C_4^2._{292}D_4 % in TeX
G:=Group("C4^2.292D4"); // GroupNames label
G:=SmallGroup(128,1972);
// by ID
G=gap.SmallGroup(128,1972);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,100,675,248,4037,1027,124]);
// 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=d*b*d=a^2*b^-1,d*c*d=a^2*c^3>;
// generators/relations
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