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G = Dic25order 100 = 22·52

Dicyclic group

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

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

Series: Derived Chief Lower central Upper central

C1C25 — Dic25
C1C5C25C50 — Dic25
C25 — Dic25
C1C2

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

25C4
5Dic5

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+5/2ζ2522253ζ2521254ζ2519256ζ2516259ζ252425ζ25132512ζ2523252ζ2517258ζ25142511ζ2518257ζ25132512ζ2523252ζ2517258ζ25142511ζ2521254ζ2519256ζ2516259ζ252425ζ2518257ζ2522253    orthogonal lifted from D25
ρ72200-1-5/2-1+5/2-1+5/2-1-5/2ζ2516259ζ25132512ζ2518257ζ2523252ζ2522253ζ25142511ζ2519256ζ252425ζ2517258ζ2521254ζ25142511ζ2519256ζ252425ζ2517258ζ25132512ζ2518257ζ2523252ζ2522253ζ2521254ζ2516259    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+5/2ζ2517258ζ2519256ζ2516259ζ252425ζ25142511ζ2518257ζ2522253ζ25132512ζ2521254ζ2523252ζ2518257ζ2522253ζ25132512ζ2521254ζ2519256ζ2516259ζ252425ζ25142511ζ2523252ζ2517258    orthogonal lifted from D25
ρ102200-1+5/2-1-5/2-1-5/2-1+5/2ζ2518257ζ252425ζ25142511ζ2521254ζ2519256ζ2522253ζ25132512ζ2523252ζ2516259ζ2517258ζ2522253ζ25132512ζ2523252ζ2516259ζ252425ζ25142511ζ2521254ζ2519256ζ2517258ζ2518257    orthogonal lifted from D25
ρ112200-1-5/2-1+5/2-1+5/2-1-5/2ζ25142511ζ2523252ζ2522253ζ2517258ζ25132512ζ2519256ζ252425ζ2521254ζ2518257ζ2516259ζ2519256ζ252425ζ2521254ζ2518257ζ2523252ζ2522253ζ2517258ζ25132512ζ2516259ζ25142511    orthogonal lifted from D25
ρ122200-1-5/2-1+5/2-1+5/2-1-5/2ζ2521254ζ2522253ζ2517258ζ25132512ζ2518257ζ2516259ζ25142511ζ2519256ζ2523252ζ252425ζ2516259ζ25142511ζ2519256ζ2523252ζ2522253ζ2517258ζ25132512ζ2518257ζ252425ζ2521254    orthogonal lifted from D25
ρ132200-1-5/2-1+5/2-1+5/2-1-5/2ζ252425ζ2518257ζ2523252ζ2522253ζ2517258ζ2521254ζ2516259ζ25142511ζ25132512ζ2519256ζ2521254ζ2516259ζ25142511ζ25132512ζ2518257ζ2523252ζ2522253ζ2517258ζ2519256ζ252425    orthogonal lifted from D25
ρ142200-1-5/2-1+5/2-1+5/2-1-5/2ζ2519256ζ2517258ζ25132512ζ2518257ζ2523252ζ252425ζ2521254ζ2516259ζ2522253ζ25142511ζ252425ζ2521254ζ2516259ζ2522253ζ2517258ζ25132512ζ2518257ζ2523252ζ25142511ζ2519256    orthogonal lifted from D25
ρ152200-1+5/2-1-5/2-1-5/2-1+5/2ζ2523252ζ25142511ζ2521254ζ2519256ζ2516259ζ2517258ζ2518257ζ2522253ζ252425ζ25132512ζ2517258ζ2518257ζ2522253ζ252425ζ25142511ζ2521254ζ2519256ζ2516259ζ25132512ζ2523252    orthogonal lifted from D25
ρ162200-1+5/2-1-5/2-1-5/2-1+5/2ζ25132512ζ2516259ζ252425ζ25142511ζ2521254ζ2523252ζ2517258ζ2518257ζ2519256ζ2522253ζ2523252ζ2517258ζ2518257ζ2519256ζ2516259ζ252425ζ25142511ζ2521254ζ2522253ζ25132512    orthogonal lifted from D25
ρ172-200-1-5/2-1+5/21-5/21+5/2ζ2516259ζ25132512ζ2518257ζ2523252ζ2522253ζ25142511ζ2519256ζ252425ζ2517258ζ252125425142511251925625242525172582513251225182572523252252225325212542516259    symplectic faithful, Schur index 2
ρ182-200-1-5/2-1+5/21-5/21+5/2ζ2521254ζ2522253ζ2517258ζ25132512ζ2518257ζ2516259ζ25142511ζ2519256ζ2523252ζ25242525162592514251125192562523252252225325172582513251225182572524252521254    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+5/21-5/2ζ2523252ζ25142511ζ2521254ζ2519256ζ2516259ζ2517258ζ2518257ζ2522253ζ252425ζ2513251225172582518257252225325242525142511252125425192562516259251325122523252    symplectic faithful, Schur index 2
ρ212-200-1-5/2-1+5/21-5/21+5/2ζ252425ζ2518257ζ2523252ζ2522253ζ2517258ζ2521254ζ2516259ζ25142511ζ25132512ζ251925625212542516259251425112513251225182572523252252225325172582519256252425    symplectic faithful, Schur index 2
ρ222-200-1+5/2-1-5/21+5/21-5/2ζ2517258ζ2519256ζ2516259ζ252425ζ25142511ζ2518257ζ2522253ζ25132512ζ2521254ζ252325225182572522253251325122521254251925625162592524252514251125232522517258    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+5/21-5/2ζ2518257ζ252425ζ25142511ζ2521254ζ2519256ζ2522253ζ25132512ζ2523252ζ2516259ζ251725825222532513251225232522516259252425251425112521254251925625172582518257    symplectic faithful, Schur index 2
ρ252-200-1+5/2-1-5/21+5/21-5/2ζ25132512ζ2516259ζ252425ζ25142511ζ2521254ζ2523252ζ2517258ζ2518257ζ2519256ζ252225325232522517258251825725192562516259252425251425112521254252225325132512    symplectic faithful, Schur index 2
ρ262-200-1+5/2-1-5/21+5/21-5/2ζ2522253ζ2521254ζ2519256ζ2516259ζ252425ζ25132512ζ2523252ζ2517258ζ25142511ζ251825725132512252325225172582514251125212542519256251625925242525182572522253    symplectic faithful, Schur index 2
ρ272-200-1-5/2-1+5/21-5/21+5/2ζ25142511ζ2523252ζ2522253ζ2517258ζ25132512ζ2519256ζ252425ζ2521254ζ2518257ζ251625925192562524252521254251825725232522522253251725825132512251625925142511    symplectic faithful, Schur index 2
ρ282-200-1-5/2-1+5/21-5/21+5/2ζ2519256ζ2517258ζ25132512ζ2518257ζ2523252ζ252425ζ2521254ζ2516259ζ2522253ζ2514251125242525212542516259252225325172582513251225182572523252251425112519256    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

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

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