p-group, metabelian, nilpotent (class 3), monomial
Aliases: C4.192+ 1+4, C8⋊2D4⋊14C2, C8⋊8D4⋊28C2, D4⋊D4⋊25C2, (C2×D4).159D4, C8.D4⋊14C2, (C2×Q8).135D4, D4.7D4⋊26C2, C4⋊C4.143C23, (C2×C8).323C23, (C2×C4).402C24, (C2×D8).69C22, C23.283(C2×D4), C4.Q8.82C22, C2.44(D4○SD16), (C2×D4).152C23, C4⋊D4.42C22, C22⋊C8.45C22, (C2×Q8).140C23, (C2×Q16).69C22, C22⋊Q8.42C22, C23.47D4⋊11C2, C23.46D4⋊11C2, C2.83(C23⋊3D4), (C22×C8).350C22, (C22×C4).305C23, Q8⋊C4.43C22, (C2×SD16).82C22, C22.662(C22×D4), D4⋊C4.105C22, (C2×M4(2)).86C22, C22.31C24⋊10C2, (C2×C4).153(C2×D4), (C22×C8)⋊C2⋊17C2, (C2×C4⋊C4).644C22, (C2×C4○D4).170C22, SmallGroup(128,1936)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C4.192+ 1+4
G = < a,b,c,d,e | a4=1, b4=c2=e2=a2, d2=a-1b2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=a-1b3, be=eb, dcd-1=ece-1=a2c, ede-1=ab2d >
Subgroups: 444 in 201 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, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C22⋊C8, D4⋊C4, Q8⋊C4, C4.Q8, C4.Q8, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C2×C4○D4, C2×C4○D4, (C22×C8)⋊C2, D4⋊D4, D4.7D4, C8⋊8D4, C8⋊2D4, C8.D4, C23.46D4, C23.47D4, C22.31C24, C4.192+ 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C23⋊3D4, D4○SD16, C4.192+ 1+4
Character table of C4.192+ 1+4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 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 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 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 | -4 | 0 | 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 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | -2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | -2√-2 | 0 | 0 | complex lifted from D4○SD16 |
►On 64 points
Generators in S64
(1 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 25 13 29)(10 26 14 30)(11 27 15 31)(12 28 16 32)(17 52 21 56)(18 53 22 49)(19 54 23 50)(20 55 24 51)(33 43 37 47)(34 44 38 48)(35 45 39 41)(36 46 40 42)
(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 51 5 55)(2 19 6 23)(3 53 7 49)(4 21 8 17)(9 34 13 38)(10 43 14 47)(11 36 15 40)(12 45 16 41)(18 57 22 61)(20 59 24 63)(25 44 29 48)(26 37 30 33)(27 46 31 42)(28 39 32 35)(50 64 54 60)(52 58 56 62)
(1 47 61 39)(2 34 62 42)(3 45 63 37)(4 40 64 48)(5 43 57 35)(6 38 58 46)(7 41 59 33)(8 36 60 44)(9 52 31 19)(10 22 32 55)(11 50 25 17)(12 20 26 53)(13 56 27 23)(14 18 28 51)(15 54 29 21)(16 24 30 49)
(1 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 21 13 17)(10 22 14 18)(11 23 15 19)(12 24 16 20)(25 56 29 52)(26 49 30 53)(27 50 31 54)(28 51 32 55)(41 63 45 59)(42 64 46 60)(43 57 47 61)(44 58 48 62)
G:=sub<Sym(64)| (1,63,5,59)(2,64,6,60)(3,57,7,61)(4,58,8,62)(9,25,13,29)(10,26,14,30)(11,27,15,31)(12,28,16,32)(17,52,21,56)(18,53,22,49)(19,54,23,50)(20,55,24,51)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42), (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), (1,51,5,55)(2,19,6,23)(3,53,7,49)(4,21,8,17)(9,34,13,38)(10,43,14,47)(11,36,15,40)(12,45,16,41)(18,57,22,61)(20,59,24,63)(25,44,29,48)(26,37,30,33)(27,46,31,42)(28,39,32,35)(50,64,54,60)(52,58,56,62), (1,47,61,39)(2,34,62,42)(3,45,63,37)(4,40,64,48)(5,43,57,35)(6,38,58,46)(7,41,59,33)(8,36,60,44)(9,52,31,19)(10,22,32,55)(11,50,25,17)(12,20,26,53)(13,56,27,23)(14,18,28,51)(15,54,29,21)(16,24,30,49), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,21,13,17)(10,22,14,18)(11,23,15,19)(12,24,16,20)(25,56,29,52)(26,49,30,53)(27,50,31,54)(28,51,32,55)(41,63,45,59)(42,64,46,60)(43,57,47,61)(44,58,48,62)>;
G:=Group( (1,63,5,59)(2,64,6,60)(3,57,7,61)(4,58,8,62)(9,25,13,29)(10,26,14,30)(11,27,15,31)(12,28,16,32)(17,52,21,56)(18,53,22,49)(19,54,23,50)(20,55,24,51)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42), (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), (1,51,5,55)(2,19,6,23)(3,53,7,49)(4,21,8,17)(9,34,13,38)(10,43,14,47)(11,36,15,40)(12,45,16,41)(18,57,22,61)(20,59,24,63)(25,44,29,48)(26,37,30,33)(27,46,31,42)(28,39,32,35)(50,64,54,60)(52,58,56,62), (1,47,61,39)(2,34,62,42)(3,45,63,37)(4,40,64,48)(5,43,57,35)(6,38,58,46)(7,41,59,33)(8,36,60,44)(9,52,31,19)(10,22,32,55)(11,50,25,17)(12,20,26,53)(13,56,27,23)(14,18,28,51)(15,54,29,21)(16,24,30,49), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,21,13,17)(10,22,14,18)(11,23,15,19)(12,24,16,20)(25,56,29,52)(26,49,30,53)(27,50,31,54)(28,51,32,55)(41,63,45,59)(42,64,46,60)(43,57,47,61)(44,58,48,62) );
G=PermutationGroup([[(1,63,5,59),(2,64,6,60),(3,57,7,61),(4,58,8,62),(9,25,13,29),(10,26,14,30),(11,27,15,31),(12,28,16,32),(17,52,21,56),(18,53,22,49),(19,54,23,50),(20,55,24,51),(33,43,37,47),(34,44,38,48),(35,45,39,41),(36,46,40,42)], [(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)], [(1,51,5,55),(2,19,6,23),(3,53,7,49),(4,21,8,17),(9,34,13,38),(10,43,14,47),(11,36,15,40),(12,45,16,41),(18,57,22,61),(20,59,24,63),(25,44,29,48),(26,37,30,33),(27,46,31,42),(28,39,32,35),(50,64,54,60),(52,58,56,62)], [(1,47,61,39),(2,34,62,42),(3,45,63,37),(4,40,64,48),(5,43,57,35),(6,38,58,46),(7,41,59,33),(8,36,60,44),(9,52,31,19),(10,22,32,55),(11,50,25,17),(12,20,26,53),(13,56,27,23),(14,18,28,51),(15,54,29,21),(16,24,30,49)], [(1,35,5,39),(2,36,6,40),(3,37,7,33),(4,38,8,34),(9,21,13,17),(10,22,14,18),(11,23,15,19),(12,24,16,20),(25,56,29,52),(26,49,30,53),(27,50,31,54),(28,51,32,55),(41,63,45,59),(42,64,46,60),(43,57,47,61),(44,58,48,62)]])
Matrix representation of C4.192+ 1+4 ►in GL8(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 16 | 2 | 0 | 0 | 0 | 0 |
| 1 | 0 | 16 | 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 |
| 5 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 5 | 10 | 7 | 0 | 0 | 0 | 0 |
| 12 | 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 9 | 9 | 8 |
| 0 | 0 | 0 | 0 | 8 | 8 | 9 | 9 |
| 0 | 0 | 0 | 0 | 9 | 8 | 9 | 8 |
| 0 | 0 | 0 | 0 | 9 | 9 | 9 | 9 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 1 | 15 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 16 | 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 |
| 1 | 16 | 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 5 |
| 0 | 0 | 0 | 0 | 12 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 5 |
| 1 | 16 | 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 12 |
G:=sub<GL(8,GF(17))| [0,1,1,1,0,0,0,0,16,0,16,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[5,12,12,12,0,0,0,0,5,5,5,0,0,0,0,0,0,0,10,5,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,8,8,9,9,0,0,0,0,9,8,8,9,0,0,0,0,9,9,9,9,0,0,0,0,8,9,8,9],[0,16,16,16,0,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,16,16,0,0,0,0,0,0,2,0,0,16,0,0,0,0,0,0,0,0,5,0,12,0,0,0,0,0,0,12,0,5,0,0,0,0,12,0,12,0,0,0,0,0,0,5,0,5],[1,0,0,16,0,0,0,0,16,0,16,0,0,0,0,0,16,1,0,1,0,0,0,0,2,0,0,16,0,0,0,0,0,0,0,0,5,0,12,0,0,0,0,0,0,5,0,12,0,0,0,0,12,0,12,0,0,0,0,0,0,12,0,12] >;
C4.192+ 1+4 in GAP, Magma, Sage, TeX
C_4._{19}2_+^{1+4} % in TeX
G:=Group("C4.19ES+(2,2)"); // GroupNames label
G:=SmallGroup(128,1936);
// by ID
G=gap.SmallGroup(128,1936);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,675,1018,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=1,b^4=c^2=e^2=a^2,d^2=a^-1*b^2,d*b*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^-1*b^3,b*e=e*b,d*c*d^-1=e*c*e^-1=a^2*c,e*d*e^-1=a*b^2*d>;
// generators/relations
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