Copied to
clipboard

G = Dic25  order 100 = 22·52

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic25, C25⋊2C4, 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 >

C1
C2
C5
25C4
C10
C25
5Dic5
C50
Dic25

Character table of Dic25

 class 124A4B5A5B10A10B25A25B25C25D25E25F25G25H25I25J50A50B50C50D50E50F50G50H50I50J
 size 112525222222222222222222222222
ρ11111111111111111111111111111    trivial
ρ211-1-1111111111111111111111111    linear of order 2
ρ31-1-ii11-1-11111111111-1-1-1-1-1-1-1-1-1-1    linear of order 4
ρ41-1i-i11-1-11111111111-1-1-1-1-1-1-1-1-1-1    linear of order 4
ρ522002222-1-√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
ρ62200-1+√5/2-1-√5/2-1-√5/2-1+√5/2ζ2522+ζ253ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2513+ζ2512ζ2523+ζ252ζ2517+ζ258ζ2514+ζ2511ζ2518+ζ257ζ2513+ζ2512ζ2523+ζ252ζ2517+ζ258ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2518+ζ257ζ2522+ζ253    orthogonal lifted from D25
ρ72200-1-√5/2-1+√5/2-1+√5/2-1-√5/2ζ2516+ζ259ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2514+ζ2511ζ2519+ζ256ζ2524+ζ25ζ2517+ζ258ζ2521+ζ254ζ2514+ζ2511ζ2519+ζ256ζ2524+ζ25ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2521+ζ254ζ2516+ζ259    orthogonal lifted from D25
ρ822002222-1+√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
ρ92200-1+√5/2-1-√5/2-1-√5/2-1+√5/2ζ2517+ζ258ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2518+ζ257ζ2522+ζ253ζ2513+ζ2512ζ2521+ζ254ζ2523+ζ252ζ2518+ζ257ζ2522+ζ253ζ2513+ζ2512ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2523+ζ252ζ2517+ζ258    orthogonal lifted from D25
ρ102200-1+√5/2-1-√5/2-1-√5/2-1+√5/2ζ2518+ζ257ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2522+ζ253ζ2513+ζ2512ζ2523+ζ252ζ2516+ζ259ζ2517+ζ258ζ2522+ζ253ζ2513+ζ2512ζ2523+ζ252ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2517+ζ258ζ2518+ζ257    orthogonal lifted from D25
ρ112200-1-√5/2-1+√5/2-1+√5/2-1-√5/2ζ2514+ζ2511ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2519+ζ256ζ2524+ζ25ζ2521+ζ254ζ2518+ζ257ζ2516+ζ259ζ2519+ζ256ζ2524+ζ25ζ2521+ζ254ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2516+ζ259ζ2514+ζ2511    orthogonal lifted from D25
ρ122200-1-√5/2-1+√5/2-1+√5/2-1-√5/2ζ2521+ζ254ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2516+ζ259ζ2514+ζ2511ζ2519+ζ256ζ2523+ζ252ζ2524+ζ25ζ2516+ζ259ζ2514+ζ2511ζ2519+ζ256ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2524+ζ25ζ2521+ζ254    orthogonal lifted from D25
ρ132200-1-√5/2-1+√5/2-1+√5/2-1-√5/2ζ2524+ζ25ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2521+ζ254ζ2516+ζ259ζ2514+ζ2511ζ2513+ζ2512ζ2519+ζ256ζ2521+ζ254ζ2516+ζ259ζ2514+ζ2511ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2519+ζ256ζ2524+ζ25    orthogonal lifted from D25
ρ142200-1-√5/2-1+√5/2-1+√5/2-1-√5/2ζ2519+ζ256ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2524+ζ25ζ2521+ζ254ζ2516+ζ259ζ2522+ζ253ζ2514+ζ2511ζ2524+ζ25ζ2521+ζ254ζ2516+ζ259ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2514+ζ2511ζ2519+ζ256    orthogonal lifted from D25
ρ152200-1+√5/2-1-√5/2-1-√5/2-1+√5/2ζ2523+ζ252ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2517+ζ258ζ2518+ζ257ζ2522+ζ253ζ2524+ζ25ζ2513+ζ2512ζ2517+ζ258ζ2518+ζ257ζ2522+ζ253ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2513+ζ2512ζ2523+ζ252    orthogonal lifted from D25
ρ162200-1+√5/2-1-√5/2-1-√5/2-1+√5/2ζ2513+ζ2512ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2523+ζ252ζ2517+ζ258ζ2518+ζ257ζ2519+ζ256ζ2522+ζ253ζ2523+ζ252ζ2517+ζ258ζ2518+ζ257ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2522+ζ253ζ2513+ζ2512    orthogonal lifted from D25
ρ172-200-1-√5/2-1+√5/21-√5/21+√5/2ζ2516+ζ259ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2514+ζ2511ζ2519+ζ256ζ2524+ζ25ζ2517+ζ258ζ2521+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
ρ182-200-1-√5/2-1+√5/21-√5/21+√5/2ζ2521+ζ254ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2516+ζ259ζ2514+ζ2511ζ2519+ζ256ζ2523+ζ252ζ2524+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
ρ192-20022-2-2-1+√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+√5/21-√5/21-√5/21-√5/21+√5/21+√5/21+√5/21+√5/21+√5/21-√5/21-√5/2    symplectic lifted from Dic5, Schur index 2
ρ202-200-1+√5/2-1-√5/21+√5/21-√5/2ζ2523+ζ252ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2517+ζ258ζ2518+ζ257ζ2522+ζ253ζ2524+ζ25ζ2513+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
ρ212-200-1-√5/2-1+√5/21-√5/21+√5/2ζ2524+ζ25ζ2518+ζ257ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2521+ζ254ζ2516+ζ259ζ2514+ζ2511ζ2513+ζ2512ζ2519+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
ρ222-200-1+√5/2-1-√5/21+√5/21-√5/2ζ2517+ζ258ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2518+ζ257ζ2522+ζ253ζ2513+ζ2512ζ2521+ζ254ζ2523+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
ρ232-20022-2-2-1-√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-√5/21+√5/21+√5/21+√5/21-√5/21-√5/21-√5/21-√5/21-√5/21+√5/21+√5/2    symplectic lifted from Dic5, Schur index 2
ρ242-200-1+√5/2-1-√5/21+√5/21-√5/2ζ2518+ζ257ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2519+ζ256ζ2522+ζ253ζ2513+ζ2512ζ2523+ζ252ζ2516+ζ259ζ2517+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
ρ252-200-1+√5/2-1-√5/21+√5/21-√5/2ζ2513+ζ2512ζ2516+ζ259ζ2524+ζ25ζ2514+ζ2511ζ2521+ζ254ζ2523+ζ252ζ2517+ζ258ζ2518+ζ257ζ2519+ζ256ζ2522+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
ρ262-200-1+√5/2-1-√5/21+√5/21-√5/2ζ2522+ζ253ζ2521+ζ254ζ2519+ζ256ζ2516+ζ259ζ2524+ζ25ζ2513+ζ2512ζ2523+ζ252ζ2517+ζ258ζ2514+ζ2511ζ2518+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
ρ272-200-1-√5/2-1+√5/21-√5/21+√5/2ζ2514+ζ2511ζ2523+ζ252ζ2522+ζ253ζ2517+ζ258ζ2513+ζ2512ζ2519+ζ256ζ2524+ζ25ζ2521+ζ254ζ2518+ζ257ζ2516+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
ρ282-200-1-√5/2-1+√5/21-√5/21+√5/2ζ2519+ζ256ζ2517+ζ258ζ2513+ζ2512ζ2518+ζ257ζ2523+ζ252ζ2524+ζ25ζ2521+ζ254ζ2516+ζ259ζ2522+ζ253ζ2514+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 90 26 65)(2 89 27 64)(3 88 28 63)(4 87 29 62)(5 86 30 61)(6 85 31 60)(7 84 32 59)(8 83 33 58)(9 82 34 57)(10 81 35 56)(11 80 36 55)(12 79 37 54)(13 78 38 53)(14 77 39 52)(15 76 40 51)(16 75 41 100)(17 74 42 99)(18 73 43 98)(19 72 44 97)(20 71 45 96)(21 70 46 95)(22 69 47 94)(23 68 48 93)(24 67 49 92)(25 66 50 91)
 
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), (1,90,26,65)(2,89,27,64)(3,88,28,63)(4,87,29,62)(5,86,30,61)(6,85,31,60)(7,84,32,59)(8,83,33,58)(9,82,34,57)(10,81,35,56)(11,80,36,55)(12,79,37,54)(13,78,38,53)(14,77,39,52)(15,76,40,51)(16,75,41,100)(17,74,42,99)(18,73,43,98)(19,72,44,97)(20,71,45,96)(21,70,46,95)(22,69,47,94)(23,68,48,93)(24,67,49,92)(25,66,50,91)>;
 
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), (1,90,26,65)(2,89,27,64)(3,88,28,63)(4,87,29,62)(5,86,30,61)(6,85,31,60)(7,84,32,59)(8,83,33,58)(9,82,34,57)(10,81,35,56)(11,80,36,55)(12,79,37,54)(13,78,38,53)(14,77,39,52)(15,76,40,51)(16,75,41,100)(17,74,42,99)(18,73,43,98)(19,72,44,97)(20,71,45,96)(21,70,46,95)(22,69,47,94)(23,68,48,93)(24,67,49,92)(25,66,50,91) );
 
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)], [(1,90,26,65),(2,89,27,64),(3,88,28,63),(4,87,29,62),(5,86,30,61),(6,85,31,60),(7,84,32,59),(8,83,33,58),(9,82,34,57),(10,81,35,56),(11,80,36,55),(12,79,37,54),(13,78,38,53),(14,77,39,52),(15,76,40,51),(16,75,41,100),(17,74,42,99),(18,73,43,98),(19,72,44,97),(20,71,45,96),(21,70,46,95),(22,69,47,94),(23,68,48,93),(24,67,49,92),(25,66,50,91)]])
 

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
 C25⋊2C8  Dic75  Dic125  C50.D5  D5.D25

Matrix representation of Dic25 ►in GL3(𝔽101) generated by

10000
0698
09343
,
9100
06070
07741
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
 

Export

Subgroup lattice of Dic25 in TeX
Character table of Dic25 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁