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## G = D5×D15order 300 = 22·3·52

### Direct product of D5 and D15

Aliases: D5×D15, C51D30, C523D6, C153D10, C31D52, (C5×D5)⋊S3, (C3×D5)⋊D5, C52(S3×D5), C5⋊D152C2, (C5×D15)⋊2C2, (D5×C15)⋊1C2, (C5×C15)⋊3C22, SmallGroup(300,39)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C15 — D5×D15
 Chief series C1 — C5 — C52 — C5×C15 — C5×D15 — D5×D15
 Lower central C5×C15 — D5×D15
 Upper central C1

Generators and relations for D5×D15
G = < a,b,c,d | a5=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

5C2
15C2
75C2
2C5
2C5
75C22
5C6
5S3
25S3
3D5
5C10
15C10
15D5
15D5
30D5
30D5
2C15
2C15
25D6
15D10
15D10
5C30
5D15
5D15
10D15
10D15
5D30
3D52

Permutation representations of D5×D15
On 30 points - transitive group 30T67
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 28)(2 29)(3 30)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)

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

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

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

G:=TransitiveGroup(30,67);

36 conjugacy classes

 class 1 2A 2B 2C 3 5A 5B 5C 5D 5E 5F 5G 5H 6 10A 10B 10C 10D 15A 15B 15C 15D 15E ··· 15N 30A 30B 30C 30D order 1 2 2 2 3 5 5 5 5 5 5 5 5 6 10 10 10 10 15 15 15 15 15 ··· 15 30 30 30 30 size 1 5 15 75 2 2 2 2 2 4 4 4 4 10 10 10 30 30 2 2 2 2 4 ··· 4 10 10 10 10

36 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D5 D5 D6 D10 D15 D30 S3×D5 D52 D5×D15 kernel D5×D15 D5×C15 C5×D15 C5⋊D15 C5×D5 C3×D5 D15 C52 C15 D5 C5 C5 C3 C1 # reps 1 1 1 1 1 2 2 1 4 4 4 2 4 8

Matrix representation of D5×D15 in GL4(𝔽31) generated by

 1 0 0 0 0 1 0 0 0 0 18 30 0 0 1 0
,
 30 0 0 0 0 30 0 0 0 0 18 30 0 0 13 13
,
 26 7 0 0 3 8 0 0 0 0 1 0 0 0 0 1
,
 0 15 0 0 29 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(31))| [1,0,0,0,0,1,0,0,0,0,18,1,0,0,30,0],[30,0,0,0,0,30,0,0,0,0,18,13,0,0,30,13],[26,3,0,0,7,8,0,0,0,0,1,0,0,0,0,1],[0,29,0,0,15,0,0,0,0,0,1,0,0,0,0,1] >;

D5×D15 in GAP, Magma, Sage, TeX

D_5\times D_{15}
% in TeX

G:=Group("D5xD15");
// GroupNames label

G:=SmallGroup(300,39);
// by ID

G=gap.SmallGroup(300,39);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5,122,488,6004]);
// Polycyclic

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

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