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## G = D5×C15order 150 = 2·3·52

### Direct product of C15 and D5

Aliases: D5×C15, C5⋊C30, C523C6, C152C10, (C5×C15)⋊3C2, SmallGroup(150,8)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D5×C15
G = < a,b,c | a15=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Permutation representations of D5×C15
On 30 points - transitive group 30T39
Generators in S30
(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 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 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 16)(11 17)(12 18)(13 19)(14 20)(15 21)

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

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

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

G:=TransitiveGroup(30,39);

D5×C15 is a maximal subgroup of   D5.D15

60 conjugacy classes

 class 1 2 3A 3B 5A 5B 5C 5D 5E ··· 5N 6A 6B 10A 10B 10C 10D 15A ··· 15H 15I ··· 15AB 30A ··· 30H order 1 2 3 3 5 5 5 5 5 ··· 5 6 6 10 10 10 10 15 ··· 15 15 ··· 15 30 ··· 30 size 1 5 1 1 1 1 1 1 2 ··· 2 5 5 5 5 5 5 1 ··· 1 2 ··· 2 5 ··· 5

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C5 C6 C10 C15 C30 D5 C3×D5 C5×D5 D5×C15 kernel D5×C15 C5×C15 C5×D5 C3×D5 C52 C15 D5 C5 C15 C5 C3 C1 # reps 1 1 2 4 2 4 8 8 2 4 8 16

Matrix representation of D5×C15 in GL2(𝔽31) generated by

 18 0 0 18
,
 16 0 3 2
,
 29 30 3 2
G:=sub<GL(2,GF(31))| [18,0,0,18],[16,3,0,2],[29,3,30,2] >;

D5×C15 in GAP, Magma, Sage, TeX

D_5\times C_{15}
% in TeX

G:=Group("D5xC15");
// GroupNames label

G:=SmallGroup(150,8);
// by ID

G=gap.SmallGroup(150,8);
# by ID

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

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

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