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G = C5×S3≀C2  order 360 = 23·32·5

Direct product of C5 and S3≀C2

direct product, non-abelian, soluble, monomial

Aliases: C5×S3≀C2, S32⋊C10, C32⋊(C5×D4), C32⋊C4⋊C10, (C3×C15)⋊3D4, (C5×S32)⋊2C2, (C5×C32⋊C4)⋊3C2, C3⋊S3.1(C2×C10), (C5×C3⋊S3).4C22, SmallGroup(360,132)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3⋊S3 — C5×S3≀C2
Chief series
C1 — C32 — C3⋊S3 — C5×C3⋊S3 — C5×S32 — C5×S3≀C2
Lower central
C32 — C3⋊S3 — C5×S3≀C2
Upper central
C1 — C5

Generators and relations for C5×S3≀C2
 G = < a,b,c,d,e | a5=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

C1
6C2
6C2
9C2
2C3
2C3
C5
9C22
9C22
9C4
2S3
2S3
6S3
6C6
6S3
6C6
C32
6C10
6C10
9C10
2C15
2C15
9D4
6D6
6D6
C3⋊S3
2C3×S3
2C3×S3
9C2×C10
9C2×C10
9C20
2C5×S3
2C5×S3
6C30
6C30
6C5×S3
6C5×S3
C3×C15
S32
C32⋊C4
S32
9C5×D4
6S3×C10
6S3×C10
C5×C3⋊S3
2S3×C15
2S3×C15
S3≀C2
C5×S32
C5×C32⋊C4
C5×S32
C5×S3≀C2

Permutation representations of C5×S3≀C2
►On 30 points - transitive group 30T82
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)

(6 29 15)(7 30 11)(8 26 12)(9 27 13)(10 28 14)

(1 23 17)(2 24 18)(3 25 19)(4 21 20)(5 22 16)

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

(11 30)(12 26)(13 27)(14 28)(15 29)
 
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), (6,29,15)(7,30,11)(8,26,12)(9,27,13)(10,28,14), (1,23,17)(2,24,18)(3,25,19)(4,21,20)(5,22,16), (1,8)(2,9)(3,10)(4,6)(5,7)(11,22,30,16)(12,23,26,17)(13,24,27,18)(14,25,28,19)(15,21,29,20), (11,30)(12,26)(13,27)(14,28)(15,29)>;
 
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), (6,29,15)(7,30,11)(8,26,12)(9,27,13)(10,28,14), (1,23,17)(2,24,18)(3,25,19)(4,21,20)(5,22,16), (1,8)(2,9)(3,10)(4,6)(5,7)(11,22,30,16)(12,23,26,17)(13,24,27,18)(14,25,28,19)(15,21,29,20), (11,30)(12,26)(13,27)(14,28)(15,29) );
 
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)], [(6,29,15),(7,30,11),(8,26,12),(9,27,13),(10,28,14)], [(1,23,17),(2,24,18),(3,25,19),(4,21,20),(5,22,16)], [(1,8),(2,9),(3,10),(4,6),(5,7),(11,22,30,16),(12,23,26,17),(13,24,27,18),(14,25,28,19),(15,21,29,20)], [(11,30),(12,26),(13,27),(14,28),(15,29)]])
 
G:=TransitiveGroup(30,82);
 

45 conjugacy classes

class 1 2A2B2C3A3B 4 5A5B5C5D6A6B10A···10H10I10J10K10L15A···15H20A20B20C20D30A···30H
order122233455556610···101010101015···152020202030···30
size16694418111112126···699994···41818181812···12

45 irreducible representations

dim1111112244
type+++++
imageC1C2C2C5C10C10D4C5×D4S3≀C2C5×S3≀C2
kernelC5×S3≀C2C5×C32⋊C4C5×S32S3≀C2C32⋊C4S32C3×C15C32C5C1
# reps11244814416

Matrix representation of C5×S3≀C2 ►in GL6(𝔽61)

5800000
0580000
001000
000100
000010
000001
,
100000
010000
0006000
0016000
000101
006006060
,
100000
010000
0006000
0016000
006006060
000110
,
0600000
100000
0000601
0060605960
000010
000110
,
0600000
6000000
0000601
0060605960
000010
001010

G:=sub<GL(6,GF(61))| [58,0,0,0,0,0,0,58,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,60,0,0,60,60,1,0,0,0,0,0,0,60,0,0,0,0,1,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,60,0,0,0,60,60,0,1,0,0,0,0,60,1,0,0,0,0,60,0],[0,1,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,60,0,1,0,0,60,59,1,1,0,0,1,60,0,0],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,1,0,0,0,60,0,0,0,0,60,59,1,1,0,0,1,60,0,0] >;
 

C5×S3≀C2 in GAP, Magma, Sage, TeX 

C_5\times S_3\wr C_2
 
% in TeX
 
G:=Group("C5xS3wrC2");
 
// GroupNames label
 
G:=SmallGroup(360,132);
 
// by ID
 
G=gap.SmallGroup(360,132);
 
# by ID
 
G:=PCGroup([6,-2,-2,-5,-2,-3,3,265,2404,1810,142,731,455]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^5=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C5×S3≀C2 in TeX

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