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G = Dic25order 100 = 22·52

Dicyclic group

Aliases: Dic25, C252C4, C50.C2, C2.D25, C5.Dic5, C10.1D5, SmallGroup(100,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — Dic25
 Chief series C1 — C5 — C25 — C50 — Dic25
 Lower central C25 — Dic25
 Upper central C1 — C2

Generators and relations for Dic25
G = < a,b | a50=1, b2=a25, bab-1=a-1 >

Character table of Dic25

 class 1 2 4A 4B 5A 5B 10A 10B 25A 25B 25C 25D 25E 25F 25G 25H 25I 25J 50A 50B 50C 50D 50E 50F 50G 50H 50I 50J size 1 1 25 25 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 linear of order 2 ρ3 1 -1 -i i 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 4 ρ4 1 -1 i -i 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 4 ρ5 2 2 0 0 2 2 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ6 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 ζ2522+ζ253 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2513+ζ2512 ζ2523+ζ252 ζ2517+ζ258 ζ2514+ζ2511 ζ2518+ζ257 ζ2513+ζ2512 ζ2523+ζ252 ζ2517+ζ258 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2518+ζ257 ζ2522+ζ253 orthogonal lifted from D25 ρ7 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ2516+ζ259 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2514+ζ2511 ζ2519+ζ256 ζ2524+ζ25 ζ2517+ζ258 ζ2521+ζ254 ζ2514+ζ2511 ζ2519+ζ256 ζ2524+ζ25 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2521+ζ254 ζ2516+ζ259 orthogonal lifted from D25 ρ8 2 2 0 0 2 2 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ9 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 ζ2517+ζ258 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2518+ζ257 ζ2522+ζ253 ζ2513+ζ2512 ζ2521+ζ254 ζ2523+ζ252 ζ2518+ζ257 ζ2522+ζ253 ζ2513+ζ2512 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2523+ζ252 ζ2517+ζ258 orthogonal lifted from D25 ρ10 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 ζ2518+ζ257 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2522+ζ253 ζ2513+ζ2512 ζ2523+ζ252 ζ2516+ζ259 ζ2517+ζ258 ζ2522+ζ253 ζ2513+ζ2512 ζ2523+ζ252 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2517+ζ258 ζ2518+ζ257 orthogonal lifted from D25 ρ11 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ2514+ζ2511 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2519+ζ256 ζ2524+ζ25 ζ2521+ζ254 ζ2518+ζ257 ζ2516+ζ259 ζ2519+ζ256 ζ2524+ζ25 ζ2521+ζ254 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2516+ζ259 ζ2514+ζ2511 orthogonal lifted from D25 ρ12 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ2521+ζ254 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2516+ζ259 ζ2514+ζ2511 ζ2519+ζ256 ζ2523+ζ252 ζ2524+ζ25 ζ2516+ζ259 ζ2514+ζ2511 ζ2519+ζ256 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2524+ζ25 ζ2521+ζ254 orthogonal lifted from D25 ρ13 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ2524+ζ25 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2521+ζ254 ζ2516+ζ259 ζ2514+ζ2511 ζ2513+ζ2512 ζ2519+ζ256 ζ2521+ζ254 ζ2516+ζ259 ζ2514+ζ2511 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2519+ζ256 ζ2524+ζ25 orthogonal lifted from D25 ρ14 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 ζ2519+ζ256 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2524+ζ25 ζ2521+ζ254 ζ2516+ζ259 ζ2522+ζ253 ζ2514+ζ2511 ζ2524+ζ25 ζ2521+ζ254 ζ2516+ζ259 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2514+ζ2511 ζ2519+ζ256 orthogonal lifted from D25 ρ15 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 ζ2523+ζ252 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2517+ζ258 ζ2518+ζ257 ζ2522+ζ253 ζ2524+ζ25 ζ2513+ζ2512 ζ2517+ζ258 ζ2518+ζ257 ζ2522+ζ253 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2513+ζ2512 ζ2523+ζ252 orthogonal lifted from D25 ρ16 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 ζ2513+ζ2512 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2523+ζ252 ζ2517+ζ258 ζ2518+ζ257 ζ2519+ζ256 ζ2522+ζ253 ζ2523+ζ252 ζ2517+ζ258 ζ2518+ζ257 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2522+ζ253 ζ2513+ζ2512 orthogonal lifted from D25 ρ17 2 -2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 ζ2516+ζ259 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2514+ζ2511 ζ2519+ζ256 ζ2524+ζ25 ζ2517+ζ258 ζ2521+ζ254 -ζ2514-ζ2511 -ζ2519-ζ256 -ζ2524-ζ25 -ζ2517-ζ258 -ζ2513-ζ2512 -ζ2518-ζ257 -ζ2523-ζ252 -ζ2522-ζ253 -ζ2521-ζ254 -ζ2516-ζ259 symplectic faithful, Schur index 2 ρ18 2 -2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 ζ2521+ζ254 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2516+ζ259 ζ2514+ζ2511 ζ2519+ζ256 ζ2523+ζ252 ζ2524+ζ25 -ζ2516-ζ259 -ζ2514-ζ2511 -ζ2519-ζ256 -ζ2523-ζ252 -ζ2522-ζ253 -ζ2517-ζ258 -ζ2513-ζ2512 -ζ2518-ζ257 -ζ2524-ζ25 -ζ2521-ζ254 symplectic faithful, Schur index 2 ρ19 2 -2 0 0 2 2 -2 -2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ20 2 -2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 ζ2523+ζ252 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2517+ζ258 ζ2518+ζ257 ζ2522+ζ253 ζ2524+ζ25 ζ2513+ζ2512 -ζ2517-ζ258 -ζ2518-ζ257 -ζ2522-ζ253 -ζ2524-ζ25 -ζ2514-ζ2511 -ζ2521-ζ254 -ζ2519-ζ256 -ζ2516-ζ259 -ζ2513-ζ2512 -ζ2523-ζ252 symplectic faithful, Schur index 2 ρ21 2 -2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 ζ2524+ζ25 ζ2518+ζ257 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2521+ζ254 ζ2516+ζ259 ζ2514+ζ2511 ζ2513+ζ2512 ζ2519+ζ256 -ζ2521-ζ254 -ζ2516-ζ259 -ζ2514-ζ2511 -ζ2513-ζ2512 -ζ2518-ζ257 -ζ2523-ζ252 -ζ2522-ζ253 -ζ2517-ζ258 -ζ2519-ζ256 -ζ2524-ζ25 symplectic faithful, Schur index 2 ρ22 2 -2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 ζ2517+ζ258 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2518+ζ257 ζ2522+ζ253 ζ2513+ζ2512 ζ2521+ζ254 ζ2523+ζ252 -ζ2518-ζ257 -ζ2522-ζ253 -ζ2513-ζ2512 -ζ2521-ζ254 -ζ2519-ζ256 -ζ2516-ζ259 -ζ2524-ζ25 -ζ2514-ζ2511 -ζ2523-ζ252 -ζ2517-ζ258 symplectic faithful, Schur index 2 ρ23 2 -2 0 0 2 2 -2 -2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ24 2 -2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 ζ2518+ζ257 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2519+ζ256 ζ2522+ζ253 ζ2513+ζ2512 ζ2523+ζ252 ζ2516+ζ259 ζ2517+ζ258 -ζ2522-ζ253 -ζ2513-ζ2512 -ζ2523-ζ252 -ζ2516-ζ259 -ζ2524-ζ25 -ζ2514-ζ2511 -ζ2521-ζ254 -ζ2519-ζ256 -ζ2517-ζ258 -ζ2518-ζ257 symplectic faithful, Schur index 2 ρ25 2 -2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 ζ2513+ζ2512 ζ2516+ζ259 ζ2524+ζ25 ζ2514+ζ2511 ζ2521+ζ254 ζ2523+ζ252 ζ2517+ζ258 ζ2518+ζ257 ζ2519+ζ256 ζ2522+ζ253 -ζ2523-ζ252 -ζ2517-ζ258 -ζ2518-ζ257 -ζ2519-ζ256 -ζ2516-ζ259 -ζ2524-ζ25 -ζ2514-ζ2511 -ζ2521-ζ254 -ζ2522-ζ253 -ζ2513-ζ2512 symplectic faithful, Schur index 2 ρ26 2 -2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 ζ2522+ζ253 ζ2521+ζ254 ζ2519+ζ256 ζ2516+ζ259 ζ2524+ζ25 ζ2513+ζ2512 ζ2523+ζ252 ζ2517+ζ258 ζ2514+ζ2511 ζ2518+ζ257 -ζ2513-ζ2512 -ζ2523-ζ252 -ζ2517-ζ258 -ζ2514-ζ2511 -ζ2521-ζ254 -ζ2519-ζ256 -ζ2516-ζ259 -ζ2524-ζ25 -ζ2518-ζ257 -ζ2522-ζ253 symplectic faithful, Schur index 2 ρ27 2 -2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 ζ2514+ζ2511 ζ2523+ζ252 ζ2522+ζ253 ζ2517+ζ258 ζ2513+ζ2512 ζ2519+ζ256 ζ2524+ζ25 ζ2521+ζ254 ζ2518+ζ257 ζ2516+ζ259 -ζ2519-ζ256 -ζ2524-ζ25 -ζ2521-ζ254 -ζ2518-ζ257 -ζ2523-ζ252 -ζ2522-ζ253 -ζ2517-ζ258 -ζ2513-ζ2512 -ζ2516-ζ259 -ζ2514-ζ2511 symplectic faithful, Schur index 2 ρ28 2 -2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 ζ2519+ζ256 ζ2517+ζ258 ζ2513+ζ2512 ζ2518+ζ257 ζ2523+ζ252 ζ2524+ζ25 ζ2521+ζ254 ζ2516+ζ259 ζ2522+ζ253 ζ2514+ζ2511 -ζ2524-ζ25 -ζ2521-ζ254 -ζ2516-ζ259 -ζ2522-ζ253 -ζ2517-ζ258 -ζ2513-ζ2512 -ζ2518-ζ257 -ζ2523-ζ252 -ζ2514-ζ2511 -ζ2519-ζ256 symplectic faithful, Schur index 2

Smallest permutation representation of Dic25
Regular action on 100 points
Generators in S100
(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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)
(1 100 26 75)(2 99 27 74)(3 98 28 73)(4 97 29 72)(5 96 30 71)(6 95 31 70)(7 94 32 69)(8 93 33 68)(9 92 34 67)(10 91 35 66)(11 90 36 65)(12 89 37 64)(13 88 38 63)(14 87 39 62)(15 86 40 61)(16 85 41 60)(17 84 42 59)(18 83 43 58)(19 82 44 57)(20 81 45 56)(21 80 46 55)(22 79 47 54)(23 78 48 53)(24 77 49 52)(25 76 50 51)

G:=sub<Sym(100)| (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,100,26,75)(2,99,27,74)(3,98,28,73)(4,97,29,72)(5,96,30,71)(6,95,31,70)(7,94,32,69)(8,93,33,68)(9,92,34,67)(10,91,35,66)(11,90,36,65)(12,89,37,64)(13,88,38,63)(14,87,39,62)(15,86,40,61)(16,85,41,60)(17,84,42,59)(18,83,43,58)(19,82,44,57)(20,81,45,56)(21,80,46,55)(22,79,47,54)(23,78,48,53)(24,77,49,52)(25,76,50,51)>;

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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,100,26,75)(2,99,27,74)(3,98,28,73)(4,97,29,72)(5,96,30,71)(6,95,31,70)(7,94,32,69)(8,93,33,68)(9,92,34,67)(10,91,35,66)(11,90,36,65)(12,89,37,64)(13,88,38,63)(14,87,39,62)(15,86,40,61)(16,85,41,60)(17,84,42,59)(18,83,43,58)(19,82,44,57)(20,81,45,56)(21,80,46,55)(22,79,47,54)(23,78,48,53)(24,77,49,52)(25,76,50,51) );

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),(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,100,26,75),(2,99,27,74),(3,98,28,73),(4,97,29,72),(5,96,30,71),(6,95,31,70),(7,94,32,69),(8,93,33,68),(9,92,34,67),(10,91,35,66),(11,90,36,65),(12,89,37,64),(13,88,38,63),(14,87,39,62),(15,86,40,61),(16,85,41,60),(17,84,42,59),(18,83,43,58),(19,82,44,57),(20,81,45,56),(21,80,46,55),(22,79,47,54),(23,78,48,53),(24,77,49,52),(25,76,50,51)])

Dic25 is a maximal subgroup of
C25⋊C8  Dic50  C4×D25  C25⋊D4  Dic75  Dic125  C50.C10  C50.D5  D5.D25
Dic25 is a maximal quotient of
C252C8  Dic75  Dic125  C50.D5  D5.D25

Matrix representation of Dic25 in GL3(𝔽101) generated by

 100 0 0 0 69 8 0 93 43
,
 91 0 0 0 60 70 0 77 41
G:=sub<GL(3,GF(101))| [100,0,0,0,69,93,0,8,43],[91,0,0,0,60,77,0,70,41] >;

Dic25 in GAP, Magma, Sage, TeX

{\rm Dic}_{25}
% in TeX

G:=Group("Dic25");
// GroupNames label

G:=SmallGroup(100,1);
// by ID

G=gap.SmallGroup(100,1);
# by ID

G:=PCGroup([4,-2,-2,-5,-5,8,434,250,1283]);
// Polycyclic

G:=Group<a,b|a^50=1,b^2=a^25,b*a*b^-1=a^-1>;
// generators/relations

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