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G = Dic15order 60 = 22·3·5

Dicyclic group

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

Aliases: Dic15, C6.D5, C3⋊Dic5, C153C4, C10.S3, C2.D15, C52Dic3, C30.1C2, SmallGroup(60,3)

Series: Derived Chief Lower central Upper central

C1C15 — Dic15
C1C5C15C30 — Dic15
C15 — Dic15
C1C2

Generators and relations for Dic15
 G = < a,b | a30=1, b2=a15, bab-1=a-1 >

15C4
5Dic3
3Dic5

Character table of Dic15

 class 1234A4B5A5B610A10B15A15B15C15D30A30B30C30D
 size 11215152222222222222
ρ1111111111111111111    trivial
ρ2111-1-11111111111111    linear of order 2
ρ31-11i-i11-1-1-11111-1-1-1-1    linear of order 4
ρ41-11-ii11-1-1-11111-1-1-1-1    linear of order 4
ρ522200-1-5/2-1+5/22-1+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
ρ622-10022-122-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ722-100-1+5/2-1-5/2-1-1-5/2-1+5/23ζ533ζ5253ζ3ζ543ζ55ζ3ζ533ζ5252ζ32ζ5432ζ55ζ3ζ533ζ5252ζ3ζ543ζ55ζ32ζ5432ζ553ζ533ζ5253    orthogonal lifted from D15
ρ822-100-1-5/2-1+5/2-1-1+5/2-1-5/2ζ32ζ5432ζ553ζ533ζ5253ζ3ζ543ζ55ζ3ζ533ζ5252ζ3ζ543ζ553ζ533ζ5253ζ3ζ533ζ5252ζ32ζ5432ζ55    orthogonal lifted from D15
ρ922200-1+5/2-1-5/22-1-5/2-1+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
ρ1022-100-1-5/2-1+5/2-1-1+5/2-1-5/2ζ3ζ543ζ55ζ3ζ533ζ5252ζ32ζ5432ζ553ζ533ζ5253ζ32ζ5432ζ55ζ3ζ533ζ52523ζ533ζ5253ζ3ζ543ζ55    orthogonal lifted from D15
ρ1122-100-1+5/2-1-5/2-1-1-5/2-1+5/2ζ3ζ533ζ5252ζ32ζ5432ζ553ζ533ζ5253ζ3ζ543ζ553ζ533ζ5253ζ32ζ5432ζ55ζ3ζ543ζ55ζ3ζ533ζ5252    orthogonal lifted from D15
ρ122-2200-1-5/2-1+5/2-21-5/21+5/2-1+5/2-1-5/2-1+5/2-1-5/21-5/21+5/21+5/21-5/2    symplectic lifted from Dic5, Schur index 2
ρ132-2-100-1-5/2-1+5/211-5/21+5/2ζ32ζ5432ζ553ζ533ζ5253ζ3ζ543ζ55ζ3ζ533ζ5252ζ32ζ5432ζ554ζ3ζ533ζ52533ζ533ζ5252ζ3ζ543ζ554    symplectic faithful, Schur index 2
ρ142-2200-1+5/2-1-5/2-21+5/21-5/2-1-5/2-1+5/2-1-5/2-1+5/21+5/21-5/21-5/21+5/2    symplectic lifted from Dic5, Schur index 2
ρ152-2-100221-2-2-1-1-1-11111    symplectic lifted from Dic3, Schur index 2
ρ162-2-100-1+5/2-1-5/211+5/21-5/23ζ533ζ5253ζ3ζ543ζ55ζ3ζ533ζ5252ζ32ζ5432ζ553ζ533ζ5252ζ32ζ5432ζ554ζ3ζ543ζ554ζ3ζ533ζ5253    symplectic faithful, Schur index 2
ρ172-2-100-1-5/2-1+5/211-5/21+5/2ζ3ζ543ζ55ζ3ζ533ζ5252ζ32ζ5432ζ553ζ533ζ5253ζ3ζ543ζ5543ζ533ζ5252ζ3ζ533ζ5253ζ32ζ5432ζ554    symplectic faithful, Schur index 2
ρ182-2-100-1+5/2-1-5/211+5/21-5/2ζ3ζ533ζ5252ζ32ζ5432ζ553ζ533ζ5253ζ3ζ543ζ55ζ3ζ533ζ5253ζ3ζ543ζ554ζ32ζ5432ζ5543ζ533ζ5252    symplectic faithful, Schur index 2

Smallest permutation representation of Dic15
Regular action on 60 points
Generators in S60
(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)
(1 37 16 52)(2 36 17 51)(3 35 18 50)(4 34 19 49)(5 33 20 48)(6 32 21 47)(7 31 22 46)(8 60 23 45)(9 59 24 44)(10 58 25 43)(11 57 26 42)(12 56 27 41)(13 55 28 40)(14 54 29 39)(15 53 30 38)

G:=sub<Sym(60)| (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), (1,37,16,52)(2,36,17,51)(3,35,18,50)(4,34,19,49)(5,33,20,48)(6,32,21,47)(7,31,22,46)(8,60,23,45)(9,59,24,44)(10,58,25,43)(11,57,26,42)(12,56,27,41)(13,55,28,40)(14,54,29,39)(15,53,30,38)>;

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), (1,37,16,52)(2,36,17,51)(3,35,18,50)(4,34,19,49)(5,33,20,48)(6,32,21,47)(7,31,22,46)(8,60,23,45)(9,59,24,44)(10,58,25,43)(11,57,26,42)(12,56,27,41)(13,55,28,40)(14,54,29,39)(15,53,30,38) );

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)], [(1,37,16,52),(2,36,17,51),(3,35,18,50),(4,34,19,49),(5,33,20,48),(6,32,21,47),(7,31,22,46),(8,60,23,45),(9,59,24,44),(10,58,25,43),(11,57,26,42),(12,56,27,41),(13,55,28,40),(14,54,29,39),(15,53,30,38)])

Matrix representation of Dic15 in GL2(𝔽29) generated by

107
75
,
128
017
G:=sub<GL(2,GF(29))| [10,7,7,5],[12,0,8,17] >;

Dic15 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}
% in TeX

G:=Group("Dic15");
// GroupNames label

G:=SmallGroup(60,3);
// by ID

G=gap.SmallGroup(60,3);
# by ID

G:=PCGroup([4,-2,-2,-3,-5,8,98,771]);
// Polycyclic

G:=Group<a,b|a^30=1,b^2=a^15,b*a*b^-1=a^-1>;
// generators/relations

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