p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.497C23, C4.912- 1+4, C4⋊C4.385D4, (C4×Q16)⋊34C2, C8⋊3Q8⋊10C2, C8⋊6D4.7C2, C4.Q16⋊40C2, Q8⋊Q8⋊24C2, (C2×D4).335D4, C8.18(C4○D4), C8.D4⋊34C2, D4⋊3Q8.7C2, C22⋊C4.68D4, C4⋊C4.261C23, C4⋊C8.129C22, (C2×C8).112C23, (C2×C4).548C24, (C4×C8).198C22, C23.353(C2×D4), C4⋊Q8.177C22, C2.95(D4○SD16), (C4×D4).188C22, C4.79(C8.C22), (C2×Q8).248C23, (C4×Q8).185C22, C2.101(D4⋊6D4), M4(2)⋊C4⋊43C2, C4.Q8.113C22, C2.D8.227C22, C23.47D4⋊24C2, C23.20D4⋊50C2, C22⋊C8.107C22, (C22×C4).348C23, Q8⋊C4.84C22, (C2×Q16).163C22, C22.808(C22×D4), C22⋊Q8.111C22, C42⋊C2.219C22, (C2×M4(2)).141C22, C22.50C24.7C2, C4.130(C2×C4○D4), (C2×C4).632(C2×D4), C2.84(C2×C8.C22), (C2×C4⋊C4).697C22, SmallGroup(128,2088)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.497C23
G = < a,b,c,d,e | a4=b4=1, c2=a2b2, d2=a2, e2=b2, ab=ba, cac-1=a-1, dad-1=ab2, eae-1=a-1b2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, ede-1=b2d >
Subgroups: 296 in 171 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C22⋊C8, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C22⋊Q8, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C2×M4(2), C2×Q16, M4(2)⋊C4, C8⋊6D4, C4×Q16, C8.D4, Q8⋊Q8, C4.Q16, C23.47D4, C23.20D4, C8⋊3Q8, D4⋊3Q8, C22.50C24, C42.497C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8.C22, C22×D4, C2×C4○D4, 2- 1+4, D4⋊6D4, C2×C8.C22, D4○SD16, C42.497C23
Character table of C42.497C23
| 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 | 0 | 0 | -2 | 2 | 0 | 2 | 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 | 0 | 0 | -2 | -2 | 0 | -2 | 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 | 0 | 0 | 2 | -2 | 0 | 2 | 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 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | -2i | -2i | 0 | 0 | 2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 2i | 2i | 0 | 0 | -2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 2i | -2i | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | -2i | 2i | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 4 | -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 |
| 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 | -2√-2 | 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 | 0 | 2√-2 | -2√-2 | 0 | 0 | 0 | complex lifted from D4○SD16 |
►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 23 19 26)(2 24 20 27)(3 21 17 28)(4 22 18 25)(5 62 12 16)(6 63 9 13)(7 64 10 14)(8 61 11 15)(29 43 36 40)(30 44 33 37)(31 41 34 38)(32 42 35 39)(45 53 49 57)(46 54 50 58)(47 55 51 59)(48 56 52 60)
(1 58 17 56)(2 57 18 55)(3 60 19 54)(4 59 20 53)(5 44 10 39)(6 43 11 38)(7 42 12 37)(8 41 9 40)(13 36 61 31)(14 35 62 30)(15 34 63 29)(16 33 64 32)(21 52 26 46)(22 51 27 45)(23 50 28 48)(24 49 25 47)
(1 44 3 42)(2 38 4 40)(5 46 7 48)(6 51 8 49)(9 47 11 45)(10 52 12 50)(13 59 15 57)(14 56 16 54)(17 39 19 37)(18 43 20 41)(21 32 23 30)(22 36 24 34)(25 29 27 31)(26 33 28 35)(53 63 55 61)(58 64 60 62)
(1 26 19 23)(2 22 20 25)(3 28 17 21)(4 24 18 27)(5 14 12 64)(6 63 9 13)(7 16 10 62)(8 61 11 15)(29 41 36 38)(30 37 33 44)(31 43 34 40)(32 39 35 42)(45 53 49 57)(46 60 50 56)(47 55 51 59)(48 58 52 54)
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), (1,23,19,26)(2,24,20,27)(3,21,17,28)(4,22,18,25)(5,62,12,16)(6,63,9,13)(7,64,10,14)(8,61,11,15)(29,43,36,40)(30,44,33,37)(31,41,34,38)(32,42,35,39)(45,53,49,57)(46,54,50,58)(47,55,51,59)(48,56,52,60), (1,58,17,56)(2,57,18,55)(3,60,19,54)(4,59,20,53)(5,44,10,39)(6,43,11,38)(7,42,12,37)(8,41,9,40)(13,36,61,31)(14,35,62,30)(15,34,63,29)(16,33,64,32)(21,52,26,46)(22,51,27,45)(23,50,28,48)(24,49,25,47), (1,44,3,42)(2,38,4,40)(5,46,7,48)(6,51,8,49)(9,47,11,45)(10,52,12,50)(13,59,15,57)(14,56,16,54)(17,39,19,37)(18,43,20,41)(21,32,23,30)(22,36,24,34)(25,29,27,31)(26,33,28,35)(53,63,55,61)(58,64,60,62), (1,26,19,23)(2,22,20,25)(3,28,17,21)(4,24,18,27)(5,14,12,64)(6,63,9,13)(7,16,10,62)(8,61,11,15)(29,41,36,38)(30,37,33,44)(31,43,34,40)(32,39,35,42)(45,53,49,57)(46,60,50,56)(47,55,51,59)(48,58,52,54)>;
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), (1,23,19,26)(2,24,20,27)(3,21,17,28)(4,22,18,25)(5,62,12,16)(6,63,9,13)(7,64,10,14)(8,61,11,15)(29,43,36,40)(30,44,33,37)(31,41,34,38)(32,42,35,39)(45,53,49,57)(46,54,50,58)(47,55,51,59)(48,56,52,60), (1,58,17,56)(2,57,18,55)(3,60,19,54)(4,59,20,53)(5,44,10,39)(6,43,11,38)(7,42,12,37)(8,41,9,40)(13,36,61,31)(14,35,62,30)(15,34,63,29)(16,33,64,32)(21,52,26,46)(22,51,27,45)(23,50,28,48)(24,49,25,47), (1,44,3,42)(2,38,4,40)(5,46,7,48)(6,51,8,49)(9,47,11,45)(10,52,12,50)(13,59,15,57)(14,56,16,54)(17,39,19,37)(18,43,20,41)(21,32,23,30)(22,36,24,34)(25,29,27,31)(26,33,28,35)(53,63,55,61)(58,64,60,62), (1,26,19,23)(2,22,20,25)(3,28,17,21)(4,24,18,27)(5,14,12,64)(6,63,9,13)(7,16,10,62)(8,61,11,15)(29,41,36,38)(30,37,33,44)(31,43,34,40)(32,39,35,42)(45,53,49,57)(46,60,50,56)(47,55,51,59)(48,58,52,54) );
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)], [(1,23,19,26),(2,24,20,27),(3,21,17,28),(4,22,18,25),(5,62,12,16),(6,63,9,13),(7,64,10,14),(8,61,11,15),(29,43,36,40),(30,44,33,37),(31,41,34,38),(32,42,35,39),(45,53,49,57),(46,54,50,58),(47,55,51,59),(48,56,52,60)], [(1,58,17,56),(2,57,18,55),(3,60,19,54),(4,59,20,53),(5,44,10,39),(6,43,11,38),(7,42,12,37),(8,41,9,40),(13,36,61,31),(14,35,62,30),(15,34,63,29),(16,33,64,32),(21,52,26,46),(22,51,27,45),(23,50,28,48),(24,49,25,47)], [(1,44,3,42),(2,38,4,40),(5,46,7,48),(6,51,8,49),(9,47,11,45),(10,52,12,50),(13,59,15,57),(14,56,16,54),(17,39,19,37),(18,43,20,41),(21,32,23,30),(22,36,24,34),(25,29,27,31),(26,33,28,35),(53,63,55,61),(58,64,60,62)], [(1,26,19,23),(2,22,20,25),(3,28,17,21),(4,24,18,27),(5,14,12,64),(6,63,9,13),(7,16,10,62),(8,61,11,15),(29,41,36,38),(30,37,33,44),(31,43,34,40),(32,39,35,42),(45,53,49,57),(46,60,50,56),(47,55,51,59),(48,58,52,54)]])
Matrix representation of C42.497C23 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 11 | 16 | 16 | 2 |
| 0 | 0 | 5 | 11 | 16 | 1 |
| 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 | 11 | 16 | 16 | 2 |
| 0 | 0 | 5 | 11 | 16 | 1 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 6 | 1 | 1 | 15 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 12 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 13 | 5 | 0 | 7 |
| 0 | 0 | 8 | 13 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 16 | 16 | 2 |
| 0 | 0 | 6 | 5 | 16 | 1 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,11,5,0,0,1,0,16,11,0,0,0,0,16,16,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,11,5,0,0,1,0,16,11,0,0,0,0,16,16,0,0,0,0,2,1],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,6,1,12,0,0,0,1,0,0,0,0,1,1,0,12,0,0,0,15,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,12,12,13,8,0,0,12,5,5,13,0,0,0,0,0,12,0,0,0,0,7,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,11,6,0,0,16,0,16,5,0,0,0,0,16,16,0,0,0,0,2,1] >;
C42.497C23 in GAP, Magma, Sage, TeX
C_4^2._{497}C_2^3 % in TeX
G:=Group("C4^2.497C2^3"); // GroupNames label
G:=SmallGroup(128,2088);
// by ID
G=gap.SmallGroup(128,2088);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,723,436,2019,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2*b^2,d^2=a^2,e^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,e*a*e^-1=a^-1*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*b^2*c,e*d*e^-1=b^2*d>;
// generators/relations
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