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G = C5×D15order 150 = 2·3·52

Direct product of C5 and D15

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C5×D15, C153D5, C522S3, C151C10, C3⋊(C5×D5), C5⋊(C5×S3), (C5×C15)⋊2C2, SmallGroup(150,11)

Series: Derived Chief Lower central Upper central

C1C15 — C5×D15
C1C5C15C5×C15 — C5×D15
C15 — C5×D15
C1C5

Generators and relations for C5×D15
 G = < a,b,c | a5=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

15C2
2C5
2C5
5S3
3D5
15C10
2C15
2C15
5C5×S3
3C5×D5

Permutation representations of C5×D15
On 30 points - transitive group 30T36
Generators in S30
(1 13 10 7 4)(2 14 11 8 5)(3 15 12 9 6)(16 19 22 25 28)(17 20 23 26 29)(18 21 24 27 30)
(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)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)

G:=sub<Sym(30)| (1,13,10,7,4)(2,14,11,8,5)(3,15,12,9,6)(16,19,22,25,28)(17,20,23,26,29)(18,21,24,27,30), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)>;

G:=Group( (1,13,10,7,4)(2,14,11,8,5)(3,15,12,9,6)(16,19,22,25,28)(17,20,23,26,29)(18,21,24,27,30), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21) );

G=PermutationGroup([(1,13,10,7,4),(2,14,11,8,5),(3,15,12,9,6),(16,19,22,25,28),(17,20,23,26,29),(18,21,24,27,30)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21)])

G:=TransitiveGroup(30,36);

C5×D15 is a maximal subgroup of   C5×S3×D5  D15⋊D5  C52⋊D9  C52⋊(C3⋊S3)

45 conjugacy classes

class 1  2  3 5A5B5C5D5E···5N10A10B10C10D15A···15X
order12355555···51010101015···15
size115211112···2151515152···2

45 irreducible representations

dim1111222222
type+++++
imageC1C2C5C10S3D5C5×S3D15C5×D5C5×D15
kernelC5×D15C5×C15D15C15C52C15C5C5C3C1
# reps11441244816

Matrix representation of C5×D15 in GL2(𝔽31) generated by

80
08
,
70
299
,
229
299
G:=sub<GL(2,GF(31))| [8,0,0,8],[7,29,0,9],[22,29,9,9] >;

C5×D15 in GAP, Magma, Sage, TeX

C_5\times D_{15}
% in TeX

G:=Group("C5xD15");
// GroupNames label

G:=SmallGroup(150,11);
// by ID

G=gap.SmallGroup(150,11);
# by ID

G:=PCGroup([4,-2,-5,-3,-5,242,1923]);
// Polycyclic

G:=Group<a,b,c|a^5=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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Subgroup lattice of C5×D15 in TeX

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