p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.493C23, C4.872- 1+4, C4⋊C4.170D4, (C4×Q16)⋊33C2, C8⋊6D4.6C2, Q8.Q8⋊45C2, (C2×D4).184D4, C8.16(C4○D4), C8.D4⋊33C2, C2.59(Q8○D8), C8.5Q8⋊22C2, C4⋊C4.257C23, C4⋊C8.125C22, (C2×C4).544C24, (C2×C8).202C23, (C4×C8).194C22, C22⋊C4.180D4, C23.349(C2×D4), C2.97(D4⋊6D4), C4.Q8.71C22, C2.D8.66C22, (C4×D4).184C22, (C2×Q8).246C23, (C4×Q8).183C22, M4(2)⋊C4⋊39C2, C23.48D4⋊35C2, C23.20D4⋊49C2, C22⋊C8.103C22, (C22×C4).344C23, Q8⋊C4.83C22, (C2×Q16).162C22, C22.804(C22×D4), C22⋊Q8.109C22, C42.C2.57C22, C2.99(D8⋊C22), C42⋊C2.215C22, (C2×M4(2)).137C22, C22.46C24.5C2, C4.126(C2×C4○D4), (C2×C4).628(C2×D4), (C2×C4⋊C4).693C22, SmallGroup(128,2084)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.493C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2b2, d2=a2, ab=ba, cac-1=a-1b2, dad-1=ab2, eae=a-1, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, ede=b2d >
Subgroups: 280 in 168 conjugacy classes, 86 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4×C8, C22⋊C8, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, C2×M4(2), C2×Q16, M4(2)⋊C4, C8⋊6D4, C4×Q16, C8.D4, Q8.Q8, C23.48D4, C23.20D4, C8.5Q8, C22.46C24, C42.493C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, D4⋊6D4, D8⋊C22, Q8○D8, C42.493C23
Character table of C42.493C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | -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 | -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 | -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 | -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 | -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 | -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 | -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 | -1 | -1 | 1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 2i | -2i | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | -2i | 2i | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | -2i | -2i | 0 | 0 | 2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 2i | 2i | 0 | 0 | -2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 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 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ27 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ28 | 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 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ29 | 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 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►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 20 27 21)(2 17 28 22)(3 18 25 23)(4 19 26 24)(5 13 64 11)(6 14 61 12)(7 15 62 9)(8 16 63 10)(29 34 44 37)(30 35 41 38)(31 36 42 39)(32 33 43 40)(45 51 58 56)(46 52 59 53)(47 49 60 54)(48 50 57 55)
(1 46 25 57)(2 58 26 47)(3 48 27 59)(4 60 28 45)(5 44 62 31)(6 32 63 41)(7 42 64 29)(8 30 61 43)(9 39 13 34)(10 35 14 40)(11 37 15 36)(12 33 16 38)(17 51 24 54)(18 55 21 52)(19 49 22 56)(20 53 23 50)
(1 41 3 43)(2 31 4 29)(5 49 7 51)(6 55 8 53)(9 58 11 60)(10 46 12 48)(13 47 15 45)(14 57 16 59)(17 39 19 37)(18 33 20 35)(21 38 23 40)(22 36 24 34)(25 32 27 30)(26 44 28 42)(50 63 52 61)(54 62 56 64)
(2 4)(5 64)(6 63)(7 62)(8 61)(9 15)(10 14)(11 13)(12 16)(17 19)(22 24)(26 28)(29 42)(30 41)(31 44)(32 43)(33 40)(34 39)(35 38)(36 37)(46 48)(50 52)(53 55)(57 59)
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), (1,20,27,21)(2,17,28,22)(3,18,25,23)(4,19,26,24)(5,13,64,11)(6,14,61,12)(7,15,62,9)(8,16,63,10)(29,34,44,37)(30,35,41,38)(31,36,42,39)(32,33,43,40)(45,51,58,56)(46,52,59,53)(47,49,60,54)(48,50,57,55), (1,46,25,57)(2,58,26,47)(3,48,27,59)(4,60,28,45)(5,44,62,31)(6,32,63,41)(7,42,64,29)(8,30,61,43)(9,39,13,34)(10,35,14,40)(11,37,15,36)(12,33,16,38)(17,51,24,54)(18,55,21,52)(19,49,22,56)(20,53,23,50), (1,41,3,43)(2,31,4,29)(5,49,7,51)(6,55,8,53)(9,58,11,60)(10,46,12,48)(13,47,15,45)(14,57,16,59)(17,39,19,37)(18,33,20,35)(21,38,23,40)(22,36,24,34)(25,32,27,30)(26,44,28,42)(50,63,52,61)(54,62,56,64), (2,4)(5,64)(6,63)(7,62)(8,61)(9,15)(10,14)(11,13)(12,16)(17,19)(22,24)(26,28)(29,42)(30,41)(31,44)(32,43)(33,40)(34,39)(35,38)(36,37)(46,48)(50,52)(53,55)(57,59)>;
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), (1,20,27,21)(2,17,28,22)(3,18,25,23)(4,19,26,24)(5,13,64,11)(6,14,61,12)(7,15,62,9)(8,16,63,10)(29,34,44,37)(30,35,41,38)(31,36,42,39)(32,33,43,40)(45,51,58,56)(46,52,59,53)(47,49,60,54)(48,50,57,55), (1,46,25,57)(2,58,26,47)(3,48,27,59)(4,60,28,45)(5,44,62,31)(6,32,63,41)(7,42,64,29)(8,30,61,43)(9,39,13,34)(10,35,14,40)(11,37,15,36)(12,33,16,38)(17,51,24,54)(18,55,21,52)(19,49,22,56)(20,53,23,50), (1,41,3,43)(2,31,4,29)(5,49,7,51)(6,55,8,53)(9,58,11,60)(10,46,12,48)(13,47,15,45)(14,57,16,59)(17,39,19,37)(18,33,20,35)(21,38,23,40)(22,36,24,34)(25,32,27,30)(26,44,28,42)(50,63,52,61)(54,62,56,64), (2,4)(5,64)(6,63)(7,62)(8,61)(9,15)(10,14)(11,13)(12,16)(17,19)(22,24)(26,28)(29,42)(30,41)(31,44)(32,43)(33,40)(34,39)(35,38)(36,37)(46,48)(50,52)(53,55)(57,59) );
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)], [(1,20,27,21),(2,17,28,22),(3,18,25,23),(4,19,26,24),(5,13,64,11),(6,14,61,12),(7,15,62,9),(8,16,63,10),(29,34,44,37),(30,35,41,38),(31,36,42,39),(32,33,43,40),(45,51,58,56),(46,52,59,53),(47,49,60,54),(48,50,57,55)], [(1,46,25,57),(2,58,26,47),(3,48,27,59),(4,60,28,45),(5,44,62,31),(6,32,63,41),(7,42,64,29),(8,30,61,43),(9,39,13,34),(10,35,14,40),(11,37,15,36),(12,33,16,38),(17,51,24,54),(18,55,21,52),(19,49,22,56),(20,53,23,50)], [(1,41,3,43),(2,31,4,29),(5,49,7,51),(6,55,8,53),(9,58,11,60),(10,46,12,48),(13,47,15,45),(14,57,16,59),(17,39,19,37),(18,33,20,35),(21,38,23,40),(22,36,24,34),(25,32,27,30),(26,44,28,42),(50,63,52,61),(54,62,56,64)], [(2,4),(5,64),(6,63),(7,62),(8,61),(9,15),(10,14),(11,13),(12,16),(17,19),(22,24),(26,28),(29,42),(30,41),(31,44),(32,43),(33,40),(34,39),(35,38),(36,37),(46,48),(50,52),(53,55),(57,59)]])
Matrix representation of C42.493C23 ►in GL6(𝔽17)
| 1 | 2 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 4 | 8 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 16 |
| 0 | 0 | 0 | 0 | 16 | 10 |
| 0 | 0 | 10 | 1 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 7 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 0 | 0 | 16 | 7 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,16,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,8,13,0,0,0,0,0,0,0,0,10,1,0,0,0,0,1,7,0,0,7,16,0,0,0,0,16,10,0,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,7,0,0,0,0,7,1,0,0,16,7,0,0,0,0,7,1,0,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;
C42.493C23 in GAP, Magma, Sage, TeX
C_4^2._{493}C_2^3 % in TeX
G:=Group("C4^2.493C2^3"); // GroupNames label
G:=SmallGroup(128,2084);
// by ID
G=gap.SmallGroup(128,2084);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,100,2019,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2*b^2,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,e*a*e=a^-1,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=a^2*c,e*d*e=b^2*d>;
// generators/relations
Export