p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.495C23, C4.892- 1+4, (C4×D8)⋊33C2, C8⋊5(C4○D4), C8⋊3Q8⋊9C2, C8⋊2D4⋊33C2, C8⋊6D4⋊23C2, C4⋊C4.384D4, D4⋊6D4⋊16C2, D4⋊Q8⋊41C2, D4⋊2Q8⋊23C2, (C2×D4).334D4, C22⋊C4.67D4, C4⋊C4.259C23, C4⋊C8.127C22, C4.49(C8⋊C22), (C2×C8).111C23, (C4×C8).196C22, (C2×C4).546C24, C23.351(C2×D4), C4⋊Q8.176C22, C2.99(D4⋊6D4), C2.94(D4○SD16), (C2×D8).166C22, (C4×D4).186C22, (C2×D4).262C23, M4(2)⋊C4⋊41C2, C2.D8.226C22, C4.Q8.112C22, D4⋊C4.85C22, C4⋊D4.111C22, C23.19D4⋊49C2, C23.46D4⋊24C2, C22⋊C8.105C22, (C22×C4).346C23, C22.806(C22×D4), C22.49C24⋊10C2, C42⋊C2.217C22, (C2×M4(2)).139C22, C4.128(C2×C4○D4), (C2×C4).630(C2×D4), C2.85(C2×C8⋊C22), (C2×C4⋊C4).695C22, SmallGroup(128,2086)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.495C23
G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=a2b2, 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: 392 in 193 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, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C2×M4(2), C2×D8, C2×C4○D4, M4(2)⋊C4, C8⋊6D4, C4×D8, C8⋊2D4, D4⋊Q8, D4⋊2Q8, C23.46D4, C23.19D4, C8⋊3Q8, D4⋊6D4, C22.49C24, C42.495C23
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, C42.495C23
Character table of C42.495C23
| 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 | 2 | -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 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | -2 | -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 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 2 | 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 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 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 | 0 | 0 | -2 | 2 | 0 | 2i | 0 | 0 | 0 | -2i | 2i | -2i | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -2i | 0 | 0 | 0 | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2i | 0 | 0 | 0 | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -2i | 0 | 0 | 0 | 2i | -2i | 2i | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 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 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ27 | 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 | 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 | 0 | 2√-2 | -2√-2 | 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 | 0 | -2√-2 | 2√-2 | 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 52 26 31)(2 49 27 32)(3 50 28 29)(4 51 25 30)(5 47 61 53)(6 48 62 54)(7 45 63 55)(8 46 64 56)(9 34 14 39)(10 35 15 40)(11 36 16 37)(12 33 13 38)(17 57 24 41)(18 58 21 42)(19 59 22 43)(20 60 23 44)
(1 45 3 47)(2 48 4 46)(5 52 7 50)(6 51 8 49)(9 43 11 41)(10 42 12 44)(13 60 15 58)(14 59 16 57)(17 39 19 37)(18 38 20 40)(21 33 23 35)(22 36 24 34)(25 56 27 54)(26 55 28 53)(29 61 31 63)(30 64 32 62)
(1 25 28 2)(3 27 26 4)(5 54 63 46)(6 45 64 53)(7 56 61 48)(8 47 62 55)(9 15 16 12)(10 11 13 14)(17 44 22 58)(18 57 23 43)(19 42 24 60)(20 59 21 41)(29 32 52 51)(30 50 49 31)(33 39 40 36)(34 35 37 38)
(1 16 26 11)(2 10 27 15)(3 14 28 9)(4 12 25 13)(5 24 61 17)(6 20 62 23)(7 22 63 19)(8 18 64 21)(29 34 50 39)(30 38 51 33)(31 36 52 37)(32 40 49 35)(41 53 57 47)(42 46 58 56)(43 55 59 45)(44 48 60 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), (1,52,26,31)(2,49,27,32)(3,50,28,29)(4,51,25,30)(5,47,61,53)(6,48,62,54)(7,45,63,55)(8,46,64,56)(9,34,14,39)(10,35,15,40)(11,36,16,37)(12,33,13,38)(17,57,24,41)(18,58,21,42)(19,59,22,43)(20,60,23,44), (1,45,3,47)(2,48,4,46)(5,52,7,50)(6,51,8,49)(9,43,11,41)(10,42,12,44)(13,60,15,58)(14,59,16,57)(17,39,19,37)(18,38,20,40)(21,33,23,35)(22,36,24,34)(25,56,27,54)(26,55,28,53)(29,61,31,63)(30,64,32,62), (1,25,28,2)(3,27,26,4)(5,54,63,46)(6,45,64,53)(7,56,61,48)(8,47,62,55)(9,15,16,12)(10,11,13,14)(17,44,22,58)(18,57,23,43)(19,42,24,60)(20,59,21,41)(29,32,52,51)(30,50,49,31)(33,39,40,36)(34,35,37,38), (1,16,26,11)(2,10,27,15)(3,14,28,9)(4,12,25,13)(5,24,61,17)(6,20,62,23)(7,22,63,19)(8,18,64,21)(29,34,50,39)(30,38,51,33)(31,36,52,37)(32,40,49,35)(41,53,57,47)(42,46,58,56)(43,55,59,45)(44,48,60,54)>;
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), (1,52,26,31)(2,49,27,32)(3,50,28,29)(4,51,25,30)(5,47,61,53)(6,48,62,54)(7,45,63,55)(8,46,64,56)(9,34,14,39)(10,35,15,40)(11,36,16,37)(12,33,13,38)(17,57,24,41)(18,58,21,42)(19,59,22,43)(20,60,23,44), (1,45,3,47)(2,48,4,46)(5,52,7,50)(6,51,8,49)(9,43,11,41)(10,42,12,44)(13,60,15,58)(14,59,16,57)(17,39,19,37)(18,38,20,40)(21,33,23,35)(22,36,24,34)(25,56,27,54)(26,55,28,53)(29,61,31,63)(30,64,32,62), (1,25,28,2)(3,27,26,4)(5,54,63,46)(6,45,64,53)(7,56,61,48)(8,47,62,55)(9,15,16,12)(10,11,13,14)(17,44,22,58)(18,57,23,43)(19,42,24,60)(20,59,21,41)(29,32,52,51)(30,50,49,31)(33,39,40,36)(34,35,37,38), (1,16,26,11)(2,10,27,15)(3,14,28,9)(4,12,25,13)(5,24,61,17)(6,20,62,23)(7,22,63,19)(8,18,64,21)(29,34,50,39)(30,38,51,33)(31,36,52,37)(32,40,49,35)(41,53,57,47)(42,46,58,56)(43,55,59,45)(44,48,60,54) );
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)], [(1,52,26,31),(2,49,27,32),(3,50,28,29),(4,51,25,30),(5,47,61,53),(6,48,62,54),(7,45,63,55),(8,46,64,56),(9,34,14,39),(10,35,15,40),(11,36,16,37),(12,33,13,38),(17,57,24,41),(18,58,21,42),(19,59,22,43),(20,60,23,44)], [(1,45,3,47),(2,48,4,46),(5,52,7,50),(6,51,8,49),(9,43,11,41),(10,42,12,44),(13,60,15,58),(14,59,16,57),(17,39,19,37),(18,38,20,40),(21,33,23,35),(22,36,24,34),(25,56,27,54),(26,55,28,53),(29,61,31,63),(30,64,32,62)], [(1,25,28,2),(3,27,26,4),(5,54,63,46),(6,45,64,53),(7,56,61,48),(8,47,62,55),(9,15,16,12),(10,11,13,14),(17,44,22,58),(18,57,23,43),(19,42,24,60),(20,59,21,41),(29,32,52,51),(30,50,49,31),(33,39,40,36),(34,35,37,38)], [(1,16,26,11),(2,10,27,15),(3,14,28,9),(4,12,25,13),(5,24,61,17),(6,20,62,23),(7,22,63,19),(8,18,64,21),(29,34,50,39),(30,38,51,33),(31,36,52,37),(32,40,49,35),(41,53,57,47),(42,46,58,56),(43,55,59,45),(44,48,60,54)]])
Matrix representation of C42.495C23 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 1 | 0 | 1 | 16 |
| 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 | 1 | 16 | 1 | 15 |
| 0 | 0 | 1 | 0 | 1 | 16 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 9 | 7 | 10 |
| 0 | 0 | 9 | 8 | 0 | 7 |
| 0 | 0 | 1 | 9 | 9 | 8 |
| 0 | 0 | 8 | 1 | 8 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 16 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 1 | 15 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 16 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,16,1,1,0,0,1,0,16,0,0,0,0,0,1,1,0,0,0,0,15,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,1,1,0,0,1,0,16,0,0,0,0,0,1,1,0,0,0,0,15,16],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,16,9,1,8,0,0,9,8,9,1,0,0,7,0,9,8,0,0,10,7,8,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,16,0,0,0,16,0,1,0,0,0,0,0,16,0,0,0,0,0,2,1],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,1,0,0,16,0,1,0,0,0,1,16,0,1,0,0,15,0,0,16] >;
C42.495C23 in GAP, Magma, Sage, TeX
C_4^2._{495}C_2^3 % in TeX
G:=Group("C4^2.495C2^3"); // GroupNames label
G:=SmallGroup(128,2086);
// by ID
G=gap.SmallGroup(128,2086);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,100,2019,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,d^2=a^2*b^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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