p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.496C23, C4.902- 1+4, (C4×D8)⋊34C2, C8⋊6D4⋊24C2, C8⋊2D4⋊34C2, C4⋊C4.171D4, D4.Q8⋊45C2, (C2×D4).185D4, C2.59(D4○D8), C8.17(C4○D4), C8.5Q8⋊23C2, C4⋊C8.128C22, C4⋊C4.260C23, (C2×C4).547C24, (C4×C8).197C22, (C2×C8).203C23, C22⋊C4.181D4, C23.352(C2×D4), C2.D8.68C22, C4.Q8.72C22, (C2×D8).167C22, (C4×D4).187C22, (C2×D4).263C23, C22.D8⋊35C2, C2.100(D4⋊6D4), M4(2)⋊C4⋊42C2, D4⋊C4.86C22, C4⋊D4.112C22, C23.19D4⋊50C2, C22⋊C8.106C22, (C22×C4).347C23, C22.807(C22×D4), C42.C2.58C22, C2.100(D8⋊C22), C22.47C24⋊11C2, C42⋊C2.218C22, (C2×M4(2)).140C22, C4.129(C2×C4○D4), (C2×C4).631(C2×D4), (C2×C4⋊C4).696C22, SmallGroup(128,2087)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.496C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=a2b2, 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: 360 in 180 conjugacy classes, 86 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C4×C8, C22⋊C8, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C2×M4(2), C2×D8, M4(2)⋊C4, C8⋊6D4, C4×D8, C8⋊2D4, D4.Q8, C22.D8, C23.19D4, C8.5Q8, C22.47C24, C42.496C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, D4⋊6D4, D8⋊C22, D4○D8, C42.496C23
Character table of C42.496C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2i | 0 | 0 | -2i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2i | 0 | 0 | 2i | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2i | 0 | 0 | -2i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2i | 0 | 0 | 2i | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | 0 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ27 | 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 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ28 | 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 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ29 | 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 | 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 25 31 33)(2 26 32 34)(3 27 29 35)(4 28 30 36)(5 60 44 19)(6 57 41 20)(7 58 42 17)(8 59 43 18)(9 15 51 37)(10 16 52 38)(11 13 49 39)(12 14 50 40)(21 62 46 54)(22 63 47 55)(23 64 48 56)(24 61 45 53)
(1 63 3 61)(2 54 4 56)(5 52 7 50)(6 9 8 11)(10 42 12 44)(13 20 15 18)(14 60 16 58)(17 40 19 38)(21 36 23 34)(22 27 24 25)(26 46 28 48)(29 53 31 55)(30 64 32 62)(33 47 35 45)(37 59 39 57)(41 51 43 49)
(1 30 29 2)(3 32 31 4)(5 59 42 20)(6 19 43 58)(7 57 44 18)(8 17 41 60)(9 52 49 12)(10 11 50 51)(13 14 37 38)(15 16 39 40)(21 55 48 61)(22 64 45 54)(23 53 46 63)(24 62 47 56)(25 28 35 34)(26 33 36 27)
(1 10)(2 9)(3 12)(4 11)(5 55)(6 54)(7 53)(8 56)(13 28)(14 27)(15 26)(16 25)(17 45)(18 48)(19 47)(20 46)(21 57)(22 60)(23 59)(24 58)(29 50)(30 49)(31 52)(32 51)(33 38)(34 37)(35 40)(36 39)(41 62)(42 61)(43 64)(44 63)
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), (1,25,31,33)(2,26,32,34)(3,27,29,35)(4,28,30,36)(5,60,44,19)(6,57,41,20)(7,58,42,17)(8,59,43,18)(9,15,51,37)(10,16,52,38)(11,13,49,39)(12,14,50,40)(21,62,46,54)(22,63,47,55)(23,64,48,56)(24,61,45,53), (1,63,3,61)(2,54,4,56)(5,52,7,50)(6,9,8,11)(10,42,12,44)(13,20,15,18)(14,60,16,58)(17,40,19,38)(21,36,23,34)(22,27,24,25)(26,46,28,48)(29,53,31,55)(30,64,32,62)(33,47,35,45)(37,59,39,57)(41,51,43,49), (1,30,29,2)(3,32,31,4)(5,59,42,20)(6,19,43,58)(7,57,44,18)(8,17,41,60)(9,52,49,12)(10,11,50,51)(13,14,37,38)(15,16,39,40)(21,55,48,61)(22,64,45,54)(23,53,46,63)(24,62,47,56)(25,28,35,34)(26,33,36,27), (1,10)(2,9)(3,12)(4,11)(5,55)(6,54)(7,53)(8,56)(13,28)(14,27)(15,26)(16,25)(17,45)(18,48)(19,47)(20,46)(21,57)(22,60)(23,59)(24,58)(29,50)(30,49)(31,52)(32,51)(33,38)(34,37)(35,40)(36,39)(41,62)(42,61)(43,64)(44,63)>;
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), (1,25,31,33)(2,26,32,34)(3,27,29,35)(4,28,30,36)(5,60,44,19)(6,57,41,20)(7,58,42,17)(8,59,43,18)(9,15,51,37)(10,16,52,38)(11,13,49,39)(12,14,50,40)(21,62,46,54)(22,63,47,55)(23,64,48,56)(24,61,45,53), (1,63,3,61)(2,54,4,56)(5,52,7,50)(6,9,8,11)(10,42,12,44)(13,20,15,18)(14,60,16,58)(17,40,19,38)(21,36,23,34)(22,27,24,25)(26,46,28,48)(29,53,31,55)(30,64,32,62)(33,47,35,45)(37,59,39,57)(41,51,43,49), (1,30,29,2)(3,32,31,4)(5,59,42,20)(6,19,43,58)(7,57,44,18)(8,17,41,60)(9,52,49,12)(10,11,50,51)(13,14,37,38)(15,16,39,40)(21,55,48,61)(22,64,45,54)(23,53,46,63)(24,62,47,56)(25,28,35,34)(26,33,36,27), (1,10)(2,9)(3,12)(4,11)(5,55)(6,54)(7,53)(8,56)(13,28)(14,27)(15,26)(16,25)(17,45)(18,48)(19,47)(20,46)(21,57)(22,60)(23,59)(24,58)(29,50)(30,49)(31,52)(32,51)(33,38)(34,37)(35,40)(36,39)(41,62)(42,61)(43,64)(44,63) );
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)], [(1,25,31,33),(2,26,32,34),(3,27,29,35),(4,28,30,36),(5,60,44,19),(6,57,41,20),(7,58,42,17),(8,59,43,18),(9,15,51,37),(10,16,52,38),(11,13,49,39),(12,14,50,40),(21,62,46,54),(22,63,47,55),(23,64,48,56),(24,61,45,53)], [(1,63,3,61),(2,54,4,56),(5,52,7,50),(6,9,8,11),(10,42,12,44),(13,20,15,18),(14,60,16,58),(17,40,19,38),(21,36,23,34),(22,27,24,25),(26,46,28,48),(29,53,31,55),(30,64,32,62),(33,47,35,45),(37,59,39,57),(41,51,43,49)], [(1,30,29,2),(3,32,31,4),(5,59,42,20),(6,19,43,58),(7,57,44,18),(8,17,41,60),(9,52,49,12),(10,11,50,51),(13,14,37,38),(15,16,39,40),(21,55,48,61),(22,64,45,54),(23,53,46,63),(24,62,47,56),(25,28,35,34),(26,33,36,27)], [(1,10),(2,9),(3,12),(4,11),(5,55),(6,54),(7,53),(8,56),(13,28),(14,27),(15,26),(16,25),(17,45),(18,48),(19,47),(20,46),(21,57),(22,60),(23,59),(24,58),(29,50),(30,49),(31,52),(32,51),(33,38),(34,37),(35,40),(36,39),(41,62),(42,61),(43,64),(44,63)]])
Matrix representation of C42.496C23 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 4 | 9 |
| 0 | 0 | 4 | 0 | 4 | 13 |
| 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 | 1 | 16 | 1 | 15 |
| 0 | 0 | 1 | 0 | 1 | 16 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 11 | 13 | 4 |
| 0 | 0 | 11 | 6 | 0 | 13 |
| 0 | 0 | 10 | 11 | 16 | 1 |
| 0 | 0 | 6 | 5 | 6 | 5 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 4 | 9 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 4 | 9 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 | 4 | 13 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,13,4,4,0,0,4,0,13,0,0,0,0,0,4,4,0,0,0,0,9,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,1,1,0,0,1,0,16,0,0,0,0,0,1,1,0,0,0,0,15,16],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,7,11,10,6,0,0,11,6,11,5,0,0,13,0,16,6,0,0,4,13,1,5],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,4,0,0,0,4,0,13,0,0,0,0,0,4,0,0,0,0,0,9,13],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,4,0,0,4,0,0,13,0,4,0,0,0,4,13,0,4,0,0,9,0,0,13] >;
C42.496C23 in GAP, Magma, Sage, TeX
C_4^2._{496}C_2^3 % in TeX
G:=Group("C4^2.496C2^3"); // GroupNames label
G:=SmallGroup(128,2087);
// by ID
G=gap.SmallGroup(128,2087);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,436,2019,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=a^2*b^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