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## G = C5×S3≀C2order 360 = 23·32·5

### Direct product of C5 and S3≀C2

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 >

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 2A 2B 2C 3A 3B 4 5A 5B 5C 5D 6A 6B 10A ··· 10H 10I 10J 10K 10L 15A ··· 15H 20A 20B 20C 20D 30A ··· 30H order 1 2 2 2 3 3 4 5 5 5 5 6 6 10 ··· 10 10 10 10 10 15 ··· 15 20 20 20 20 30 ··· 30 size 1 6 6 9 4 4 18 1 1 1 1 12 12 6 ··· 6 9 9 9 9 4 ··· 4 18 18 18 18 12 ··· 12

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 type + + + + + image C1 C2 C2 C5 C10 C10 D4 C5×D4 S3≀C2 C5×S3≀C2 kernel C5×S3≀C2 C5×C32⋊C4 C5×S32 S3≀C2 C32⋊C4 S32 C3×C15 C32 C5 C1 # reps 1 1 2 4 4 8 1 4 4 16

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

 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 60 0 0 0 0 1 60 0 0 0 0 0 1 0 1 0 0 60 0 60 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 1 60 0 0 0 0 60 0 60 60 0 0 0 1 1 0
,
 0 60 0 0 0 0 1 0 0 0 0 0 0 0 0 0 60 1 0 0 60 60 59 60 0 0 0 0 1 0 0 0 0 1 1 0
,
 0 60 0 0 0 0 60 0 0 0 0 0 0 0 0 0 60 1 0 0 60 60 59 60 0 0 0 0 1 0 0 0 1 0 1 0

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

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