p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.63C23, C4.842- 1+4, C8⋊Q8⋊29C2, C4⋊C4.381D4, C8⋊9D4.3C2, Q8.Q8⋊42C2, C4.Q16⋊39C2, (C2×D4).181D4, C8.36(C4○D4), C8.D4⋊32C2, C2.58(Q8○D8), Q16⋊C4⋊26C2, C8.18D4⋊32C2, C4⋊C4.254C23, C4⋊C8.122C22, (C2×C8).108C23, (C2×C4).541C24, C22⋊C4.178D4, C23.346(C2×D4), C4⋊Q8.173C22, C2.94(D4⋊6D4), C8⋊C4.55C22, C4.Q8.69C22, (C4×D4).181C22, (C2×Q16).89C22, (C4×Q8).180C22, (C2×Q8).243C23, M4(2)⋊C4⋊36C2, C2.D8.197C22, C23.20D4⋊46C2, C23.48D4⋊34C2, C23.25D4⋊32C2, C22⋊C8.100C22, (C22×C8).292C22, Q8⋊C4.80C22, C22.801(C22×D4), C22⋊Q8.106C22, C42.C2.54C22, C2.96(D8⋊C22), (C22×C4).1169C23, C42⋊C2.212C22, (C2×M4(2)).134C22, C22.50C24.6C2, C22.46C24.4C2, C4.123(C2×C4○D4), (C2×C4).625(C2×D4), (C2×C4⋊C4).690C22, SmallGroup(128,2081)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.63C23
G = < a,b,c,d,e | a4=b4=1, c2=d2=a2, e2=b2, ab=ba, cac-1=eae-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2c, ede-1=b2d >
Subgroups: 280 in 168 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×Q16, C23.25D4, M4(2)⋊C4, C8⋊9D4, Q16⋊C4, C8.18D4, C8.D4, C4.Q16, Q8.Q8, C23.48D4, C23.20D4, C8⋊Q8, C22.46C24, C22.50C24, C42.63C23
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, C42.63C23
Character table of C42.63C23
| 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 | 2 | 0 | -2 | 2 | 0 | 0 | 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 | 2 | 0 | 2 | -2 | 0 | 0 | 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 | -2 | 0 | -2 | 2 | 0 | 0 | 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 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2i | 0 | 0 | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | -2i | 0 | 0 | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2i | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | -2i | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 0 | -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 | 0 | 2√2 | -2√2 | 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 | 0 | -2√2 | 2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -4i | 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 | 0 | 4i | 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 26 19 23)(2 27 20 24)(3 28 17 21)(4 25 18 22)(5 16 10 61)(6 13 11 62)(7 14 12 63)(8 15 9 64)(29 44 40 34)(30 41 37 35)(31 42 38 36)(32 43 39 33)(45 57 56 50)(46 58 53 51)(47 59 54 52)(48 60 55 49)
(1 55 3 53)(2 54 4 56)(5 32 7 30)(6 31 8 29)(9 40 11 38)(10 39 12 37)(13 36 15 34)(14 35 16 33)(17 46 19 48)(18 45 20 47)(21 51 23 49)(22 50 24 52)(25 57 27 59)(26 60 28 58)(41 61 43 63)(42 64 44 62)
(1 29 3 31)(2 37 4 39)(5 50 7 52)(6 58 8 60)(9 49 11 51)(10 57 12 59)(13 46 15 48)(14 54 16 56)(17 38 19 40)(18 32 20 30)(21 42 23 44)(22 33 24 35)(25 43 27 41)(26 34 28 36)(45 63 47 61)(53 64 55 62)
(1 23 19 26)(2 22 20 25)(3 21 17 28)(4 24 18 27)(5 16 10 61)(6 15 11 64)(7 14 12 63)(8 13 9 62)(29 34 40 44)(30 33 37 43)(31 36 38 42)(32 35 39 41)(45 57 56 50)(46 60 53 49)(47 59 54 52)(48 58 55 51)
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), (1,26,19,23)(2,27,20,24)(3,28,17,21)(4,25,18,22)(5,16,10,61)(6,13,11,62)(7,14,12,63)(8,15,9,64)(29,44,40,34)(30,41,37,35)(31,42,38,36)(32,43,39,33)(45,57,56,50)(46,58,53,51)(47,59,54,52)(48,60,55,49), (1,55,3,53)(2,54,4,56)(5,32,7,30)(6,31,8,29)(9,40,11,38)(10,39,12,37)(13,36,15,34)(14,35,16,33)(17,46,19,48)(18,45,20,47)(21,51,23,49)(22,50,24,52)(25,57,27,59)(26,60,28,58)(41,61,43,63)(42,64,44,62), (1,29,3,31)(2,37,4,39)(5,50,7,52)(6,58,8,60)(9,49,11,51)(10,57,12,59)(13,46,15,48)(14,54,16,56)(17,38,19,40)(18,32,20,30)(21,42,23,44)(22,33,24,35)(25,43,27,41)(26,34,28,36)(45,63,47,61)(53,64,55,62), (1,23,19,26)(2,22,20,25)(3,21,17,28)(4,24,18,27)(5,16,10,61)(6,15,11,64)(7,14,12,63)(8,13,9,62)(29,34,40,44)(30,33,37,43)(31,36,38,42)(32,35,39,41)(45,57,56,50)(46,60,53,49)(47,59,54,52)(48,58,55,51)>;
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), (1,26,19,23)(2,27,20,24)(3,28,17,21)(4,25,18,22)(5,16,10,61)(6,13,11,62)(7,14,12,63)(8,15,9,64)(29,44,40,34)(30,41,37,35)(31,42,38,36)(32,43,39,33)(45,57,56,50)(46,58,53,51)(47,59,54,52)(48,60,55,49), (1,55,3,53)(2,54,4,56)(5,32,7,30)(6,31,8,29)(9,40,11,38)(10,39,12,37)(13,36,15,34)(14,35,16,33)(17,46,19,48)(18,45,20,47)(21,51,23,49)(22,50,24,52)(25,57,27,59)(26,60,28,58)(41,61,43,63)(42,64,44,62), (1,29,3,31)(2,37,4,39)(5,50,7,52)(6,58,8,60)(9,49,11,51)(10,57,12,59)(13,46,15,48)(14,54,16,56)(17,38,19,40)(18,32,20,30)(21,42,23,44)(22,33,24,35)(25,43,27,41)(26,34,28,36)(45,63,47,61)(53,64,55,62), (1,23,19,26)(2,22,20,25)(3,21,17,28)(4,24,18,27)(5,16,10,61)(6,15,11,64)(7,14,12,63)(8,13,9,62)(29,34,40,44)(30,33,37,43)(31,36,38,42)(32,35,39,41)(45,57,56,50)(46,60,53,49)(47,59,54,52)(48,58,55,51) );
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)], [(1,26,19,23),(2,27,20,24),(3,28,17,21),(4,25,18,22),(5,16,10,61),(6,13,11,62),(7,14,12,63),(8,15,9,64),(29,44,40,34),(30,41,37,35),(31,42,38,36),(32,43,39,33),(45,57,56,50),(46,58,53,51),(47,59,54,52),(48,60,55,49)], [(1,55,3,53),(2,54,4,56),(5,32,7,30),(6,31,8,29),(9,40,11,38),(10,39,12,37),(13,36,15,34),(14,35,16,33),(17,46,19,48),(18,45,20,47),(21,51,23,49),(22,50,24,52),(25,57,27,59),(26,60,28,58),(41,61,43,63),(42,64,44,62)], [(1,29,3,31),(2,37,4,39),(5,50,7,52),(6,58,8,60),(9,49,11,51),(10,57,12,59),(13,46,15,48),(14,54,16,56),(17,38,19,40),(18,32,20,30),(21,42,23,44),(22,33,24,35),(25,43,27,41),(26,34,28,36),(45,63,47,61),(53,64,55,62)], [(1,23,19,26),(2,22,20,25),(3,21,17,28),(4,24,18,27),(5,16,10,61),(6,15,11,64),(7,14,12,63),(8,13,9,62),(29,34,40,44),(30,33,37,43),(31,36,38,42),(32,35,39,41),(45,57,56,50),(46,60,53,49),(47,59,54,52),(48,58,55,51)]])
Matrix representation of C42.63C23 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 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 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 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 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 6 | 0 | 0 |
| 0 | 0 | 6 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 13 |
| 0 | 0 | 0 | 0 | 13 | 11 |
| 16 | 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 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[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],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,6,0,0,0,0,6,13,0,0,0,0,0,0,6,13,0,0,0,0,13,11],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0] >;
C42.63C23 in GAP, Magma, Sage, TeX
C_4^2._{63}C_2^3 % in TeX
G:=Group("C4^2.63C2^3"); // GroupNames label
G:=SmallGroup(128,2081);
// by ID
G=gap.SmallGroup(128,2081);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,723,100,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=d^2=a^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=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^-1=a^2*c,e*d*e^-1=b^2*d>;
// generators/relations
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