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

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

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

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

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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