p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.533C23, C4.1542+ 1+4, (C4×D8)⋊36C2, C4⋊D8⋊44C2, C8⋊3D4⋊23C2, C4⋊C4.190D4, C8⋊4Q8⋊25C2, C2.73(D4○D8), C8.24(C4○D4), (C2×Q8).145D4, D4.2D4⋊51C2, C8.12D4⋊26C2, C4⋊C8.156C22, C4⋊C4.454C23, (C2×C4).595C24, (C4×C8).213C22, (C2×C8).124C23, Q8.D4⋊51C2, SD16⋊C4⋊52C2, C8⋊C4.82C22, C2.49(Q8⋊6D4), (C4×D4).228C22, (C2×D8).170C22, (C2×D4).289C23, (C4×Q8).218C22, (C2×Q8).274C23, (C2×Q16).42C22, C2.D8.231C22, C4⋊1D4.112C22, Q8⋊C4.98C22, (C2×SD16).79C22, C4.4D4.95C22, C22.855(C22×D4), D4⋊C4.103C22, C2.110(D8⋊C22), C22.53C24⋊10C2, C4.173(C2×C4○D4), (C2×C4).659(C2×D4), SmallGroup(128,2135)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.533C23
G = < a,b,c,d,e | a4=b4=c2=1, d2=a2b2, e2=a2, ab=ba, ac=ca, dad-1=a-1b2, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=a2b2c, ece-1=bc, ede-1=b2d >
Subgroups: 408 in 190 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, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C4×D4, C4×D4, C4×Q8, C4×Q8, C22.D4, C4.4D4, C4.4D4, C4⋊1D4, C2×D8, C2×D8, C2×SD16, C2×Q16, C4×D8, SD16⋊C4, C8⋊4Q8, C4⋊D8, D4.2D4, Q8.D4, C8.12D4, C8⋊3D4, C22.53C24, C42.533C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, Q8⋊6D4, D8⋊C22, D4○D8, C42.533C23
Character table of C42.533C23
| 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 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | -2 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 2 | 2 | 0 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | -2 | -2 | 0 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 2 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2i | 0 | 0 | 2i | 0 | -2i | 0 | 0 | -2i | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2i | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 2i | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| 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 | 0 | 2√2 | -2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| 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 | 0 | -2√2 | 2√2 | 0 | 0 | orthogonal lifted from D4○D8 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | -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 | 0 | 0 | -4i | 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 20 26 23)(2 17 27 24)(3 18 28 21)(4 19 25 22)(5 12 62 15)(6 9 63 16)(7 10 64 13)(8 11 61 14)(29 33 41 40)(30 34 42 37)(31 35 43 38)(32 36 44 39)(45 49 57 56)(46 50 58 53)(47 51 59 54)(48 52 60 55)
(1 60)(2 57)(3 58)(4 59)(5 42)(6 43)(7 44)(8 41)(9 35)(10 36)(11 33)(12 34)(13 39)(14 40)(15 37)(16 38)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 47)(26 48)(27 45)(28 46)(29 61)(30 62)(31 63)(32 64)
(1 44 28 30)(2 31 25 41)(3 42 26 32)(4 29 27 43)(5 58 64 48)(6 45 61 59)(7 60 62 46)(8 47 63 57)(9 49 14 54)(10 55 15 50)(11 51 16 56)(12 53 13 52)(17 35 22 40)(18 37 23 36)(19 33 24 38)(20 39 21 34)
(1 42 3 44)(2 43 4 41)(5 50 7 52)(6 51 8 49)(9 47 11 45)(10 48 12 46)(13 60 15 58)(14 57 16 59)(17 35 19 33)(18 36 20 34)(21 39 23 37)(22 40 24 38)(25 29 27 31)(26 30 28 32)(53 64 55 62)(54 61 56 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), (1,20,26,23)(2,17,27,24)(3,18,28,21)(4,19,25,22)(5,12,62,15)(6,9,63,16)(7,10,64,13)(8,11,61,14)(29,33,41,40)(30,34,42,37)(31,35,43,38)(32,36,44,39)(45,49,57,56)(46,50,58,53)(47,51,59,54)(48,52,60,55), (1,60)(2,57)(3,58)(4,59)(5,42)(6,43)(7,44)(8,41)(9,35)(10,36)(11,33)(12,34)(13,39)(14,40)(15,37)(16,38)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,47)(26,48)(27,45)(28,46)(29,61)(30,62)(31,63)(32,64), (1,44,28,30)(2,31,25,41)(3,42,26,32)(4,29,27,43)(5,58,64,48)(6,45,61,59)(7,60,62,46)(8,47,63,57)(9,49,14,54)(10,55,15,50)(11,51,16,56)(12,53,13,52)(17,35,22,40)(18,37,23,36)(19,33,24,38)(20,39,21,34), (1,42,3,44)(2,43,4,41)(5,50,7,52)(6,51,8,49)(9,47,11,45)(10,48,12,46)(13,60,15,58)(14,57,16,59)(17,35,19,33)(18,36,20,34)(21,39,23,37)(22,40,24,38)(25,29,27,31)(26,30,28,32)(53,64,55,62)(54,61,56,63)>;
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), (1,20,26,23)(2,17,27,24)(3,18,28,21)(4,19,25,22)(5,12,62,15)(6,9,63,16)(7,10,64,13)(8,11,61,14)(29,33,41,40)(30,34,42,37)(31,35,43,38)(32,36,44,39)(45,49,57,56)(46,50,58,53)(47,51,59,54)(48,52,60,55), (1,60)(2,57)(3,58)(4,59)(5,42)(6,43)(7,44)(8,41)(9,35)(10,36)(11,33)(12,34)(13,39)(14,40)(15,37)(16,38)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,47)(26,48)(27,45)(28,46)(29,61)(30,62)(31,63)(32,64), (1,44,28,30)(2,31,25,41)(3,42,26,32)(4,29,27,43)(5,58,64,48)(6,45,61,59)(7,60,62,46)(8,47,63,57)(9,49,14,54)(10,55,15,50)(11,51,16,56)(12,53,13,52)(17,35,22,40)(18,37,23,36)(19,33,24,38)(20,39,21,34), (1,42,3,44)(2,43,4,41)(5,50,7,52)(6,51,8,49)(9,47,11,45)(10,48,12,46)(13,60,15,58)(14,57,16,59)(17,35,19,33)(18,36,20,34)(21,39,23,37)(22,40,24,38)(25,29,27,31)(26,30,28,32)(53,64,55,62)(54,61,56,63) );
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)], [(1,20,26,23),(2,17,27,24),(3,18,28,21),(4,19,25,22),(5,12,62,15),(6,9,63,16),(7,10,64,13),(8,11,61,14),(29,33,41,40),(30,34,42,37),(31,35,43,38),(32,36,44,39),(45,49,57,56),(46,50,58,53),(47,51,59,54),(48,52,60,55)], [(1,60),(2,57),(3,58),(4,59),(5,42),(6,43),(7,44),(8,41),(9,35),(10,36),(11,33),(12,34),(13,39),(14,40),(15,37),(16,38),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,47),(26,48),(27,45),(28,46),(29,61),(30,62),(31,63),(32,64)], [(1,44,28,30),(2,31,25,41),(3,42,26,32),(4,29,27,43),(5,58,64,48),(6,45,61,59),(7,60,62,46),(8,47,63,57),(9,49,14,54),(10,55,15,50),(11,51,16,56),(12,53,13,52),(17,35,22,40),(18,37,23,36),(19,33,24,38),(20,39,21,34)], [(1,42,3,44),(2,43,4,41),(5,50,7,52),(6,51,8,49),(9,47,11,45),(10,48,12,46),(13,60,15,58),(14,57,16,59),(17,35,19,33),(18,36,20,34),(21,39,23,37),(22,40,24,38),(25,29,27,31),(26,30,28,32),(53,64,55,62),(54,61,56,63)]])
Matrix representation of C42.533C23 ►in GL6(𝔽17)
| 1 | 2 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 4 | 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 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 13 | 9 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[13,4,0,0,0,0,9,4,0,0,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,14,3,0,0,0,0,14,14,0,0],[13,4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42.533C23 in GAP, Magma, Sage, TeX
C_4^2._{533}C_2^3 % in TeX
G:=Group("C4^2.533C2^3"); // GroupNames label
G:=SmallGroup(128,2135);
// by ID
G=gap.SmallGroup(128,2135);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,436,346,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=a^2*b^2,e^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d>;
// generators/relations
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