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## G = Dic15order 60 = 22·3·5

### Dicyclic group

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — Dic15
 Chief series C1 — C5 — C15 — C30 — Dic15
 Lower central C15 — Dic15
 Upper central C1 — C2

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

Character table of Dic15

 class 1 2 3 4A 4B 5A 5B 6 10A 10B 15A 15B 15C 15D 30A 30B 30C 30D size 1 1 2 15 15 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 trivial ρ2 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 i -i 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ4 1 -1 1 -i i 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ5 2 2 2 0 0 -1-√5/2 -1+√5/2 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 orthogonal lifted from D5 ρ6 2 2 -1 0 0 2 2 -1 2 2 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 -1 0 0 -1+√5/2 -1-√5/2 -1 -1-√5/2 -1+√5/2 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 ζ3ζ54-ζ3ζ5-ζ5 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 orthogonal lifted from D15 ρ8 2 2 -1 0 0 -1-√5/2 -1+√5/2 -1 -1+√5/2 -1-√5/2 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 ζ3ζ54-ζ3ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 orthogonal lifted from D15 ρ9 2 2 2 0 0 -1+√5/2 -1-√5/2 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 orthogonal lifted from D5 ρ10 2 2 -1 0 0 -1-√5/2 -1+√5/2 -1 -1+√5/2 -1-√5/2 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 orthogonal lifted from D15 ρ11 2 2 -1 0 0 -1+√5/2 -1-√5/2 -1 -1-√5/2 -1+√5/2 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ32ζ54-ζ32ζ5-ζ5 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 orthogonal lifted from D15 ρ12 2 -2 2 0 0 -1-√5/2 -1+√5/2 -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 symplectic lifted from Dic5, Schur index 2 ρ13 2 -2 -1 0 0 -1-√5/2 -1+√5/2 1 1-√5/2 1+√5/2 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5+ζ54 ζ3ζ53-ζ3ζ52+ζ53 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ54-ζ3ζ5+ζ54 symplectic faithful, Schur index 2 ρ14 2 -2 2 0 0 -1+√5/2 -1-√5/2 -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 symplectic lifted from Dic5, Schur index 2 ρ15 2 -2 -1 0 0 2 2 1 -2 -2 -1 -1 -1 -1 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ16 2 -2 -1 0 0 -1+√5/2 -1-√5/2 1 1+√5/2 1-√5/2 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52+ζ52 ζ32ζ54-ζ32ζ5+ζ54 ζ3ζ54-ζ3ζ5+ζ54 ζ3ζ53-ζ3ζ52+ζ53 symplectic faithful, Schur index 2 ρ17 2 -2 -1 0 0 -1-√5/2 -1+√5/2 1 1-√5/2 1+√5/2 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5+ζ54 -ζ3ζ53+ζ3ζ52+ζ52 ζ3ζ53-ζ3ζ52+ζ53 ζ32ζ54-ζ32ζ5+ζ54 symplectic faithful, Schur index 2 ρ18 2 -2 -1 0 0 -1+√5/2 -1-√5/2 1 1+√5/2 1-√5/2 ζ3ζ53-ζ3ζ52-ζ52 ζ32ζ54-ζ32ζ5-ζ5 -ζ3ζ53+ζ3ζ52-ζ53 ζ3ζ54-ζ3ζ5-ζ5 ζ3ζ53-ζ3ζ52+ζ53 ζ3ζ54-ζ3ζ5+ζ54 ζ32ζ54-ζ32ζ5+ζ54 -ζ3ζ53+ζ3ζ52+ζ52 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 45 16 60)(2 44 17 59)(3 43 18 58)(4 42 19 57)(5 41 20 56)(6 40 21 55)(7 39 22 54)(8 38 23 53)(9 37 24 52)(10 36 25 51)(11 35 26 50)(12 34 27 49)(13 33 28 48)(14 32 29 47)(15 31 30 46)

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

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

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

Dic15 is a maximal subgroup of
D5×Dic3  S3×Dic5  C15⋊D4  C15⋊Q8  Dic30  C4×D15  C157D4  Dic45  C3⋊Dic15  Q8.D15  A4⋊Dic5  Dic75  C30.D5  D5.D15  Dic105
Dic15 is a maximal quotient of
C153C8  Dic45  C3⋊Dic15  A4⋊Dic5  Dic75  C30.D5  D5.D15  Dic105

Matrix representation of Dic15 in GL2(𝔽29) generated by

 10 7 7 5
,
 12 8 0 17
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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