Copied to
clipboard

## G = S3×C15order 90 = 2·32·5

### Direct product of C15 and S3

Aliases: S3×C15, C3⋊C30, C153C6, C321C10, (C3×C15)⋊4C2, SmallGroup(90,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C15
 Chief series C1 — C3 — C15 — C3×C15 — S3×C15
 Lower central C3 — S3×C15
 Upper central C1 — C15

Generators and relations for S3×C15
G = < a,b,c | a15=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Permutation representations of S3×C15
On 30 points - transitive group 30T15
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 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(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), (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (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), (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (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)], [(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(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(30,15);

45 conjugacy classes

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

45 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 S3 C3×S3 C5×S3 S3×C15 kernel S3×C15 C3×C15 C5×S3 C3×S3 C15 C32 S3 C3 C15 C5 C3 C1 # reps 1 1 2 4 2 4 8 8 1 2 4 8

Matrix representation of S3×C15 in GL2(𝔽31) generated by

 10 0 0 10
,
 5 0 0 25
,
 0 1 1 0
G:=sub<GL(2,GF(31))| [10,0,0,10],[5,0,0,25],[0,1,1,0] >;

S3×C15 in GAP, Magma, Sage, TeX

S_3\times C_{15}
% in TeX

G:=Group("S3xC15");
// GroupNames label

G:=SmallGroup(90,6);
// by ID

G=gap.SmallGroup(90,6);
# by ID

G:=PCGroup([4,-2,-3,-5,-3,963]);
// Polycyclic

G:=Group<a,b,c|a^15=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

׿
×
𝔽