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## G = C15×D15order 450 = 2·32·52

### Direct product of C15 and D15

Aliases: C15×D15, C15C2, C151C30, C1523C2, C5⋊(S3×C15), C3⋊(D5×C15), C152(C5×S3), (C5×C15)⋊5S3, (C5×C15)⋊6C6, (C3×C15)⋊4D5, C153(C3×D5), C524(C3×S3), C321(C5×D5), (C3×C15)⋊2C10, SmallGroup(450,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C15×D15
 Chief series C1 — C5 — C15 — C5×C15 — C152 — C15×D15
 Lower central C15 — C15×D15
 Upper central C1 — C15

Generators and relations for C15×D15
G = < a,b,c | a15=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

Permutation representations of C15×D15
On 30 points - transitive group 30T104
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 9 2 10 3 11 4 12 5 13 6 14 7 15 8)(16 23 30 22 29 21 28 20 27 19 26 18 25 17 24)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 16)(14 17)(15 18)

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

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

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

G:=TransitiveGroup(30,104);

135 conjugacy classes

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

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C5 C6 C10 C15 C30 S3 D5 C3×S3 C5×S3 C3×D5 D15 C5×D5 S3×C15 C3×D15 D5×C15 C5×D15 C15×D15 kernel C15×D15 C152 C5×D15 C3×D15 C5×C15 C3×C15 D15 C15 C5×C15 C3×C15 C52 C15 C15 C15 C32 C5 C5 C3 C3 C1 # reps 1 1 2 4 2 4 8 8 1 2 2 4 4 4 8 8 8 16 16 32

Matrix representation of C15×D15 in GL2(𝔽31) generated by

 10 0 0 10
,
 14 0 12 20
,
 11 21 12 20
G:=sub<GL(2,GF(31))| [10,0,0,10],[14,12,0,20],[11,12,21,20] >;

C15×D15 in GAP, Magma, Sage, TeX

C_{15}\times D_{15}
% in TeX

G:=Group("C15xD15");
// GroupNames label

G:=SmallGroup(450,29);
// by ID

G=gap.SmallGroup(450,29);
# by ID

G:=PCGroup([5,-2,-3,-5,-3,-5,1203,9004]);
// Polycyclic

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

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