p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.27C23, C4.162- 1+4, C4.352+ 1+4, C8⋊8D4⋊31C2, C8⋊D4⋊24C2, C4⋊C4.142D4, Q8.Q8⋊32C2, D4.Q8⋊33C2, C4⋊SD16⋊16C2, C4⋊C8.88C22, C22⋊C4.34D4, C23.95(C2×D4), D4.D4⋊16C2, C4⋊C4.199C23, (C2×C8).341C23, (C2×C4).458C24, C4⋊Q8.129C22, C4.Q8.97C22, C2.54(D4○SD16), (C2×D4).199C23, (C4×D4).137C22, C4⋊D4.53C22, C4⋊1D4.73C22, (C2×Q8).186C23, (C4×Q8).133C22, C2.D8.115C22, C22⋊Q8.52C22, (C22×C8).352C22, Q8⋊C4.61C22, (C2×SD16).45C22, C22.718(C22×D4), C42.C2.33C22, D4⋊C4.115C22, C22.35C24⋊8C2, (C22×C4).1113C23, C42.6C22⋊15C2, (C2×M4(2)).96C22, C42⋊C2.176C22, C22.34C24.4C2, C2.77(C22.31C24), (C2×C4).582(C2×D4), SmallGroup(128,1992)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.27C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a2c, ece=bc, ede=a2d >
Subgroups: 348 in 173 conjugacy classes, 84 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C42.6C22, C4⋊SD16, D4.D4, C8⋊8D4, C8⋊D4, D4.Q8, Q8.Q8, C22.34C24, C22.35C24, C42.27C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, D4○SD16, C42.27C23
Character table of C42.27C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 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 | 0 | 0 | -2 | -2 | 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 | 0 | 0 | -2 | -2 | -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 | 0 | 0 | -2 | -2 | 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 | 0 | 0 | -2 | -2 | -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 | -4 | 4 | 0 | 0 | 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 | 4 | -4 | 0 | 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 | 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 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 22 27 18)(2 23 28 19)(3 24 25 20)(4 21 26 17)(5 62 16 10)(6 63 13 11)(7 64 14 12)(8 61 15 9)(29 43 38 36)(30 44 39 33)(31 41 40 34)(32 42 37 35)(45 57 50 56)(46 58 51 53)(47 59 52 54)(48 60 49 55)
(1 59 27 54)(2 58 28 53)(3 57 25 56)(4 60 26 55)(5 44 16 33)(6 43 13 36)(7 42 14 35)(8 41 15 34)(9 40 61 31)(10 39 62 30)(11 38 63 29)(12 37 64 32)(17 49 21 48)(18 52 22 47)(19 51 23 46)(20 50 24 45)
(1 37 27 32)(2 29 28 38)(3 39 25 30)(4 31 26 40)(5 52 16 47)(6 48 13 49)(7 50 14 45)(8 46 15 51)(9 53 61 58)(10 59 62 54)(11 55 63 60)(12 57 64 56)(17 34 21 41)(18 42 22 35)(19 36 23 43)(20 44 24 33)
(5 64)(6 61)(7 62)(8 63)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(29 31)(30 32)(33 42)(34 43)(35 44)(36 41)(37 39)(38 40)(45 57)(46 58)(47 59)(48 60)(49 55)(50 56)(51 53)(52 54)
G:=sub<Sym(64)| (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), (1,22,27,18)(2,23,28,19)(3,24,25,20)(4,21,26,17)(5,62,16,10)(6,63,13,11)(7,64,14,12)(8,61,15,9)(29,43,38,36)(30,44,39,33)(31,41,40,34)(32,42,37,35)(45,57,50,56)(46,58,51,53)(47,59,52,54)(48,60,49,55), (1,59,27,54)(2,58,28,53)(3,57,25,56)(4,60,26,55)(5,44,16,33)(6,43,13,36)(7,42,14,35)(8,41,15,34)(9,40,61,31)(10,39,62,30)(11,38,63,29)(12,37,64,32)(17,49,21,48)(18,52,22,47)(19,51,23,46)(20,50,24,45), (1,37,27,32)(2,29,28,38)(3,39,25,30)(4,31,26,40)(5,52,16,47)(6,48,13,49)(7,50,14,45)(8,46,15,51)(9,53,61,58)(10,59,62,54)(11,55,63,60)(12,57,64,56)(17,34,21,41)(18,42,22,35)(19,36,23,43)(20,44,24,33), (5,64)(6,61)(7,62)(8,63)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(29,31)(30,32)(33,42)(34,43)(35,44)(36,41)(37,39)(38,40)(45,57)(46,58)(47,59)(48,60)(49,55)(50,56)(51,53)(52,54)>;
G:=Group( (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), (1,22,27,18)(2,23,28,19)(3,24,25,20)(4,21,26,17)(5,62,16,10)(6,63,13,11)(7,64,14,12)(8,61,15,9)(29,43,38,36)(30,44,39,33)(31,41,40,34)(32,42,37,35)(45,57,50,56)(46,58,51,53)(47,59,52,54)(48,60,49,55), (1,59,27,54)(2,58,28,53)(3,57,25,56)(4,60,26,55)(5,44,16,33)(6,43,13,36)(7,42,14,35)(8,41,15,34)(9,40,61,31)(10,39,62,30)(11,38,63,29)(12,37,64,32)(17,49,21,48)(18,52,22,47)(19,51,23,46)(20,50,24,45), (1,37,27,32)(2,29,28,38)(3,39,25,30)(4,31,26,40)(5,52,16,47)(6,48,13,49)(7,50,14,45)(8,46,15,51)(9,53,61,58)(10,59,62,54)(11,55,63,60)(12,57,64,56)(17,34,21,41)(18,42,22,35)(19,36,23,43)(20,44,24,33), (5,64)(6,61)(7,62)(8,63)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(29,31)(30,32)(33,42)(34,43)(35,44)(36,41)(37,39)(38,40)(45,57)(46,58)(47,59)(48,60)(49,55)(50,56)(51,53)(52,54) );
G=PermutationGroup([[(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)], [(1,22,27,18),(2,23,28,19),(3,24,25,20),(4,21,26,17),(5,62,16,10),(6,63,13,11),(7,64,14,12),(8,61,15,9),(29,43,38,36),(30,44,39,33),(31,41,40,34),(32,42,37,35),(45,57,50,56),(46,58,51,53),(47,59,52,54),(48,60,49,55)], [(1,59,27,54),(2,58,28,53),(3,57,25,56),(4,60,26,55),(5,44,16,33),(6,43,13,36),(7,42,14,35),(8,41,15,34),(9,40,61,31),(10,39,62,30),(11,38,63,29),(12,37,64,32),(17,49,21,48),(18,52,22,47),(19,51,23,46),(20,50,24,45)], [(1,37,27,32),(2,29,28,38),(3,39,25,30),(4,31,26,40),(5,52,16,47),(6,48,13,49),(7,50,14,45),(8,46,15,51),(9,53,61,58),(10,59,62,54),(11,55,63,60),(12,57,64,56),(17,34,21,41),(18,42,22,35),(19,36,23,43),(20,44,24,33)], [(5,64),(6,61),(7,62),(8,63),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(29,31),(30,32),(33,42),(34,43),(35,44),(36,41),(37,39),(38,40),(45,57),(46,58),(47,59),(48,60),(49,55),(50,56),(51,53),(52,54)]])
Matrix representation of C42.27C23 ►in GL8(𝔽17)
| 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 | 0 | 0 |
| 0 | 0 | 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 |
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 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 |
| 0 | 6 | 0 | 9 | 0 | 0 | 0 | 0 |
| 3 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 11 | 0 | 0 | 0 | 0 |
| 13 | 0 | 14 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 5 |
| 4 | 9 | 6 | 5 | 0 | 0 | 0 | 0 |
| 4 | 13 | 6 | 11 | 0 | 0 | 0 | 0 |
| 6 | 5 | 13 | 8 | 0 | 0 | 0 | 0 |
| 6 | 11 | 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 | 0 | 4 |
| 0 | 0 | 0 | 0 | 13 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,3,0,13,0,0,0,0,6,0,9,0,0,0,0,0,0,13,0,14,0,0,0,0,9,0,11,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,5,5],[4,4,6,6,0,0,0,0,9,13,5,11,0,0,0,0,6,6,13,13,0,0,0,0,5,11,8,4,0,0,0,0,0,0,0,0,10,0,13,0,0,0,0,0,0,10,0,13,0,0,0,0,4,0,7,0,0,0,0,0,0,4,0,7],[1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16] >;
C42.27C23 in GAP, Magma, Sage, TeX
C_4^2._{27}C_2^3 % in TeX
G:=Group("C4^2.27C2^3"); // GroupNames label
G:=SmallGroup(128,1992);
// by ID
G=gap.SmallGroup(128,1992);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,675,1018,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=d^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e=b*c,e*d*e=a^2*d>;
// generators/relations
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