p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.43C23, C4.532+ 1+4, C8⋊D4⋊35C2, C8⋊9D4⋊11C2, C8⋊7D4⋊37C2, C4⋊C4.154D4, Q8⋊6(C4○D4), Q8⋊5D4⋊9C2, D4⋊6D4⋊11C2, D4⋊D4⋊43C2, C4.Q16⋊34C2, Q8⋊D4⋊20C2, (C2×D4).314D4, C4⋊C8.99C22, (C2×C8).95C23, C2.47(Q8○D8), D4.2D4⋊41C2, C4⋊C4.232C23, (C2×C4).500C24, C22⋊C4.164D4, (C2×D8).38C22, C23.474(C2×D4), C4⋊Q8.148C22, SD16⋊C4⋊31C2, C8⋊C4.40C22, (C4×D4).153C22, (C2×D4).230C23, C22.D8⋊25C2, C4⋊D4.79C22, C22⋊C8.77C22, (C2×Q8).396C23, (C4×Q8).154C22, C2.136(D4⋊5D4), C2.D8.118C22, C22⋊Q8.79C22, D4⋊C4.69C22, C23.36D4⋊15C2, C23.48D4⋊25C2, C22.11(C8⋊C22), (C22×C8).305C22, Q8⋊C4.69C22, (C2×SD16).52C22, C4.4D4.63C22, C22.760(C22×D4), (C22×C4).1144C23, (C22×Q8).341C22, C42.28C22⋊13C2, (C2×M4(2)).109C22, C4.225(C2×C4○D4), (C2×C4).597(C2×D4), C2.75(C2×C8⋊C22), (C2×Q8⋊C4)⋊31C2, (C2×C4⋊C4).665C22, (C2×C4○D4).206C22, SmallGroup(128,2040)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.43C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=b2, ab=ba, cac-1=eae=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, ede=b2d >
Subgroups: 432 in 210 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, 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), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C22×Q8, C2×C4○D4, C2×C4○D4, C2×Q8⋊C4, C23.36D4, C8⋊9D4, SD16⋊C4, Q8⋊D4, D4⋊D4, D4.2D4, C8⋊7D4, C8⋊D4, C4.Q16, C22.D8, C23.48D4, C42.28C22, D4⋊6D4, Q8⋊5D4, C42.43C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8⋊C22, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C2×C8⋊C22, Q8○D8, C42.43C23
Character table of C42.43C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 4 | 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 | 2 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | -2 | 0 | 2 | 2 | 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 | 2 | 0 | -2i | 0 | -2 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 2i | 0 | -2 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | -2i | 0 | 2 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2i | 0 | 2 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -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 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 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 | orthogonal lifted from 2+ 1+4 |
| ρ27 | 4 | -4 | -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 | 0 | 0 | orthogonal lifted from C8⋊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 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ29 | 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 | symplectic lifted from Q8○D8, Schur index 2 |
►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 23 39 33)(2 24 40 34)(3 21 37 35)(4 22 38 36)(5 62 30 26)(6 63 31 27)(7 64 32 28)(8 61 29 25)(9 13 53 59)(10 14 54 60)(11 15 55 57)(12 16 56 58)(17 46 44 51)(18 47 41 52)(19 48 42 49)(20 45 43 50)
(1 56 3 54)(2 55 4 53)(5 42 7 44)(6 41 8 43)(9 40 11 38)(10 39 12 37)(13 24 15 22)(14 23 16 21)(17 30 19 32)(18 29 20 31)(25 50 27 52)(26 49 28 51)(33 58 35 60)(34 57 36 59)(45 63 47 61)(46 62 48 64)
(1 19 39 42)(2 43 40 20)(3 17 37 44)(4 41 38 18)(5 14 30 60)(6 57 31 15)(7 16 32 58)(8 59 29 13)(9 61 53 25)(10 26 54 62)(11 63 55 27)(12 28 56 64)(21 51 35 46)(22 47 36 52)(23 49 33 48)(24 45 34 50)
(1 56)(2 55)(3 54)(4 53)(5 46)(6 45)(7 48)(8 47)(9 38)(10 37)(11 40)(12 39)(13 36)(14 35)(15 34)(16 33)(17 26)(18 25)(19 28)(20 27)(21 60)(22 59)(23 58)(24 57)(29 52)(30 51)(31 50)(32 49)(41 61)(42 64)(43 63)(44 62)
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), (1,23,39,33)(2,24,40,34)(3,21,37,35)(4,22,38,36)(5,62,30,26)(6,63,31,27)(7,64,32,28)(8,61,29,25)(9,13,53,59)(10,14,54,60)(11,15,55,57)(12,16,56,58)(17,46,44,51)(18,47,41,52)(19,48,42,49)(20,45,43,50), (1,56,3,54)(2,55,4,53)(5,42,7,44)(6,41,8,43)(9,40,11,38)(10,39,12,37)(13,24,15,22)(14,23,16,21)(17,30,19,32)(18,29,20,31)(25,50,27,52)(26,49,28,51)(33,58,35,60)(34,57,36,59)(45,63,47,61)(46,62,48,64), (1,19,39,42)(2,43,40,20)(3,17,37,44)(4,41,38,18)(5,14,30,60)(6,57,31,15)(7,16,32,58)(8,59,29,13)(9,61,53,25)(10,26,54,62)(11,63,55,27)(12,28,56,64)(21,51,35,46)(22,47,36,52)(23,49,33,48)(24,45,34,50), (1,56)(2,55)(3,54)(4,53)(5,46)(6,45)(7,48)(8,47)(9,38)(10,37)(11,40)(12,39)(13,36)(14,35)(15,34)(16,33)(17,26)(18,25)(19,28)(20,27)(21,60)(22,59)(23,58)(24,57)(29,52)(30,51)(31,50)(32,49)(41,61)(42,64)(43,63)(44,62)>;
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), (1,23,39,33)(2,24,40,34)(3,21,37,35)(4,22,38,36)(5,62,30,26)(6,63,31,27)(7,64,32,28)(8,61,29,25)(9,13,53,59)(10,14,54,60)(11,15,55,57)(12,16,56,58)(17,46,44,51)(18,47,41,52)(19,48,42,49)(20,45,43,50), (1,56,3,54)(2,55,4,53)(5,42,7,44)(6,41,8,43)(9,40,11,38)(10,39,12,37)(13,24,15,22)(14,23,16,21)(17,30,19,32)(18,29,20,31)(25,50,27,52)(26,49,28,51)(33,58,35,60)(34,57,36,59)(45,63,47,61)(46,62,48,64), (1,19,39,42)(2,43,40,20)(3,17,37,44)(4,41,38,18)(5,14,30,60)(6,57,31,15)(7,16,32,58)(8,59,29,13)(9,61,53,25)(10,26,54,62)(11,63,55,27)(12,28,56,64)(21,51,35,46)(22,47,36,52)(23,49,33,48)(24,45,34,50), (1,56)(2,55)(3,54)(4,53)(5,46)(6,45)(7,48)(8,47)(9,38)(10,37)(11,40)(12,39)(13,36)(14,35)(15,34)(16,33)(17,26)(18,25)(19,28)(20,27)(21,60)(22,59)(23,58)(24,57)(29,52)(30,51)(31,50)(32,49)(41,61)(42,64)(43,63)(44,62) );
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)], [(1,23,39,33),(2,24,40,34),(3,21,37,35),(4,22,38,36),(5,62,30,26),(6,63,31,27),(7,64,32,28),(8,61,29,25),(9,13,53,59),(10,14,54,60),(11,15,55,57),(12,16,56,58),(17,46,44,51),(18,47,41,52),(19,48,42,49),(20,45,43,50)], [(1,56,3,54),(2,55,4,53),(5,42,7,44),(6,41,8,43),(9,40,11,38),(10,39,12,37),(13,24,15,22),(14,23,16,21),(17,30,19,32),(18,29,20,31),(25,50,27,52),(26,49,28,51),(33,58,35,60),(34,57,36,59),(45,63,47,61),(46,62,48,64)], [(1,19,39,42),(2,43,40,20),(3,17,37,44),(4,41,38,18),(5,14,30,60),(6,57,31,15),(7,16,32,58),(8,59,29,13),(9,61,53,25),(10,26,54,62),(11,63,55,27),(12,28,56,64),(21,51,35,46),(22,47,36,52),(23,49,33,48),(24,45,34,50)], [(1,56),(2,55),(3,54),(4,53),(5,46),(6,45),(7,48),(8,47),(9,38),(10,37),(11,40),(12,39),(13,36),(14,35),(15,34),(16,33),(17,26),(18,25),(19,28),(20,27),(21,60),(22,59),(23,58),(24,57),(29,52),(30,51),(31,50),(32,49),(41,61),(42,64),(43,63),(44,62)]])
Matrix representation of C42.43C23 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 9 | 16 | 16 |
| 0 | 0 | 9 | 6 | 0 | 16 |
| 0 | 0 | 8 | 9 | 3 | 0 |
| 0 | 0 | 6 | 14 | 8 | 2 |
| 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 | 16 | 16 | 1 | 2 |
| 0 | 0 | 0 | 1 | 16 | 16 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 9 | 16 | 16 |
| 0 | 0 | 8 | 11 | 0 | 1 |
| 0 | 0 | 5 | 5 | 3 | 6 |
| 0 | 0 | 0 | 12 | 8 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 16 | 15 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 16 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 9 | 16 | 16 |
| 0 | 0 | 9 | 6 | 0 | 16 |
| 0 | 0 | 8 | 9 | 3 | 0 |
| 0 | 0 | 6 | 14 | 8 | 2 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,13,0,0,0,0,0,0,6,9,8,6,0,0,9,6,9,14,0,0,16,0,3,8,0,0,16,16,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,0,0,1,0,16,1,0,0,0,0,1,16,0,0,0,0,2,16],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,6,8,5,0,0,0,9,11,5,12,0,0,16,0,3,8,0,0,16,1,6,14],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,16,1,0,0,0,1,0,1,0,0,1,16,0,0,0,0,0,15,0,16],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,6,9,8,6,0,0,9,6,9,14,0,0,16,0,3,8,0,0,16,16,0,2] >;
C42.43C23 in GAP, Magma, Sage, TeX
C_4^2._{43}C_2^3 % in TeX
G:=Group("C4^2.43C2^3"); // GroupNames label
G:=SmallGroup(128,2040);
// by ID
G=gap.SmallGroup(128,2040);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,352,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1,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*c,e*d*e=b^2*d>;
// generators/relations
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