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

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

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

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

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

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

Export

Subgroup lattice of Dic15 in TeX
Character table of Dic15 in TeX

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