p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.52C23, C4.622+ 1+4, C8⋊8D4⋊52C2, C8⋊D4⋊39C2, C8⋊9D4⋊19C2, C4⋊C4.368D4, Q8.Q8⋊36C2, C4⋊SD16⋊22C2, (C2×D4).172D4, Q8⋊6D4.6C2, C22⋊C4.51D4, Q16⋊C4⋊24C2, D4.7D4⋊47C2, C4⋊C8.105C22, C4⋊C4.411C23, (C2×C8).354C23, (C2×C4).509C24, Q8.25(C4○D4), C23.326(C2×D4), C8⋊C4.46C22, C2.77(D4○SD16), (C2×D4).235C23, (C4×D4).162C22, C4⋊D4.86C22, C4⋊1D4.88C22, C22⋊C8.83C22, (C2×Q8).222C23, (C2×Q16).86C22, (C4×Q8).160C22, C2.145(D4⋊5D4), C2.D8.120C22, C4.Q8.105C22, C22⋊Q8.84C22, D4⋊C4.74C22, C23.24D4⋊32C2, C23.36D4⋊20C2, C23.19D4⋊34C2, C23.46D4⋊16C2, (C22×C8).364C22, C22.769(C22×D4), C42.C2.41C22, C2.87(D8⋊C22), (C22×C4).1153C23, C22.46C24⋊5C2, Q8⋊C4.181C22, (C2×SD16).101C22, C42.29C22⋊10C2, C42⋊C2.191C22, (C2×M4(2)).116C22, C4.234(C2×C4○D4), (C2×C4).606(C2×D4), (C2×C4⋊C4).670C22, (C2×C4○D4).213C22, SmallGroup(128,2049)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.52C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=b2, ab=ba, cac-1=eae=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2b2c, ede=b2d >
Subgroups: 400 in 196 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊1D4, C22×C8, C2×M4(2), C2×SD16, C2×Q16, C2×C4○D4, C23.24D4, C23.36D4, C8⋊9D4, Q16⋊C4, D4.7D4, C4⋊SD16, C8⋊8D4, C8⋊D4, Q8.Q8, C23.46D4, C23.19D4, C42.29C22, Q8⋊6D4, C22.46C24, C42.52C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, D8⋊C22, D4○SD16, C42.52C23
Character table of C42.52C23
| 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 | 0 | -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 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | -2 | 0 | 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 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | 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 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 0 | -2 | 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 | 0 | -2 | 2 | 0 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 2 | 0 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 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 | 4i | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -4i | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ28 | 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 | complex lifted from D4○SD16 |
| ρ29 | 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 | complex lifted from D4○SD16 |
►On 64 points
(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 45 18 42)(2 46 19 43)(3 47 20 44)(4 48 17 41)(5 33 61 39)(6 34 62 40)(7 35 63 37)(8 36 64 38)(9 21 13 50)(10 22 14 51)(11 23 15 52)(12 24 16 49)(25 56 32 58)(26 53 29 59)(27 54 30 60)(28 55 31 57)
(1 59 3 57)(2 56 4 54)(5 23 7 21)(6 51 8 49)(9 33 11 35)(10 38 12 40)(13 39 15 37)(14 36 16 34)(17 60 19 58)(18 53 20 55)(22 64 24 62)(25 48 27 46)(26 44 28 42)(29 47 31 45)(30 43 32 41)(50 61 52 63)
(1 13 18 9)(2 10 19 14)(3 15 20 11)(4 12 17 16)(5 59 61 53)(6 54 62 60)(7 57 63 55)(8 56 64 58)(21 42 50 45)(22 46 51 43)(23 44 52 47)(24 48 49 41)(25 38 32 36)(26 33 29 39)(27 40 30 34)(28 35 31 37)
(1 12)(2 15)(3 10)(4 13)(5 25)(6 31)(7 27)(8 29)(9 17)(11 19)(14 20)(16 18)(21 41)(22 47)(23 43)(24 45)(26 64)(28 62)(30 63)(32 61)(33 56)(34 57)(35 54)(36 59)(37 60)(38 53)(39 58)(40 55)(42 49)(44 51)(46 52)(48 50)
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), (1,45,18,42)(2,46,19,43)(3,47,20,44)(4,48,17,41)(5,33,61,39)(6,34,62,40)(7,35,63,37)(8,36,64,38)(9,21,13,50)(10,22,14,51)(11,23,15,52)(12,24,16,49)(25,56,32,58)(26,53,29,59)(27,54,30,60)(28,55,31,57), (1,59,3,57)(2,56,4,54)(5,23,7,21)(6,51,8,49)(9,33,11,35)(10,38,12,40)(13,39,15,37)(14,36,16,34)(17,60,19,58)(18,53,20,55)(22,64,24,62)(25,48,27,46)(26,44,28,42)(29,47,31,45)(30,43,32,41)(50,61,52,63), (1,13,18,9)(2,10,19,14)(3,15,20,11)(4,12,17,16)(5,59,61,53)(6,54,62,60)(7,57,63,55)(8,56,64,58)(21,42,50,45)(22,46,51,43)(23,44,52,47)(24,48,49,41)(25,38,32,36)(26,33,29,39)(27,40,30,34)(28,35,31,37), (1,12)(2,15)(3,10)(4,13)(5,25)(6,31)(7,27)(8,29)(9,17)(11,19)(14,20)(16,18)(21,41)(22,47)(23,43)(24,45)(26,64)(28,62)(30,63)(32,61)(33,56)(34,57)(35,54)(36,59)(37,60)(38,53)(39,58)(40,55)(42,49)(44,51)(46,52)(48,50)>;
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), (1,45,18,42)(2,46,19,43)(3,47,20,44)(4,48,17,41)(5,33,61,39)(6,34,62,40)(7,35,63,37)(8,36,64,38)(9,21,13,50)(10,22,14,51)(11,23,15,52)(12,24,16,49)(25,56,32,58)(26,53,29,59)(27,54,30,60)(28,55,31,57), (1,59,3,57)(2,56,4,54)(5,23,7,21)(6,51,8,49)(9,33,11,35)(10,38,12,40)(13,39,15,37)(14,36,16,34)(17,60,19,58)(18,53,20,55)(22,64,24,62)(25,48,27,46)(26,44,28,42)(29,47,31,45)(30,43,32,41)(50,61,52,63), (1,13,18,9)(2,10,19,14)(3,15,20,11)(4,12,17,16)(5,59,61,53)(6,54,62,60)(7,57,63,55)(8,56,64,58)(21,42,50,45)(22,46,51,43)(23,44,52,47)(24,48,49,41)(25,38,32,36)(26,33,29,39)(27,40,30,34)(28,35,31,37), (1,12)(2,15)(3,10)(4,13)(5,25)(6,31)(7,27)(8,29)(9,17)(11,19)(14,20)(16,18)(21,41)(22,47)(23,43)(24,45)(26,64)(28,62)(30,63)(32,61)(33,56)(34,57)(35,54)(36,59)(37,60)(38,53)(39,58)(40,55)(42,49)(44,51)(46,52)(48,50) );
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)], [(1,45,18,42),(2,46,19,43),(3,47,20,44),(4,48,17,41),(5,33,61,39),(6,34,62,40),(7,35,63,37),(8,36,64,38),(9,21,13,50),(10,22,14,51),(11,23,15,52),(12,24,16,49),(25,56,32,58),(26,53,29,59),(27,54,30,60),(28,55,31,57)], [(1,59,3,57),(2,56,4,54),(5,23,7,21),(6,51,8,49),(9,33,11,35),(10,38,12,40),(13,39,15,37),(14,36,16,34),(17,60,19,58),(18,53,20,55),(22,64,24,62),(25,48,27,46),(26,44,28,42),(29,47,31,45),(30,43,32,41),(50,61,52,63)], [(1,13,18,9),(2,10,19,14),(3,15,20,11),(4,12,17,16),(5,59,61,53),(6,54,62,60),(7,57,63,55),(8,56,64,58),(21,42,50,45),(22,46,51,43),(23,44,52,47),(24,48,49,41),(25,38,32,36),(26,33,29,39),(27,40,30,34),(28,35,31,37)], [(1,12),(2,15),(3,10),(4,13),(5,25),(6,31),(7,27),(8,29),(9,17),(11,19),(14,20),(16,18),(21,41),(22,47),(23,43),(24,45),(26,64),(28,62),(30,63),(32,61),(33,56),(34,57),(35,54),(36,59),(37,60),(38,53),(39,58),(40,55),(42,49),(44,51),(46,52),(48,50)]])
Matrix representation of C42.52C23 ►in GL6(𝔽17)
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 13 | 4 | 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 | 13 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 9 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 8 | 0 | 0 |
| 0 | 0 | 13 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 8 |
| 0 | 0 | 0 | 0 | 13 | 13 |
G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,4,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,9,4],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,4,13,0,0,0,0,8,13,0,0,0,0,0,0,4,13,0,0,0,0,8,13] >;
C42.52C23 in GAP, Magma, Sage, TeX
C_4^2._{52}C_2^3 % in TeX
G:=Group("C4^2.52C2^3"); // GroupNames label
G:=SmallGroup(128,2049);
// by ID
G=gap.SmallGroup(128,2049);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,352,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1*b^2,d*a*d^-1=a*b^2,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*b^2*c,e*d*e=b^2*d>;
// generators/relations
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