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## G = C5×S32order 180 = 22·32·5

### Direct product of C5, S3 and S3

Aliases: C5×S32, C155D6, C3⋊S3⋊C10, (C3×S3)⋊C10, C32⋊(C2×C10), C31(S3×C10), (S3×C15)⋊3C2, (C3×C15)⋊5C22, (C5×C3⋊S3)⋊3C2, SmallGroup(180,28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C5×S32
 Chief series C1 — C3 — C32 — C3×C15 — S3×C15 — C5×S32
 Lower central C32 — C5×S32
 Upper central C1 — C5

Generators and relations for C5×S32
G = < a,b,c,d,e | a5=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Permutation representations of C5×S32
On 30 points - transitive group 30T41
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 17 23)(2 18 24)(3 19 25)(4 20 21)(5 16 22)(6 29 15)(7 30 11)(8 26 12)(9 27 13)(10 28 14)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 16)(12 17)(13 18)(14 19)(15 20)(21 29)(22 30)(23 26)(24 27)(25 28)
(1 23 17)(2 24 18)(3 25 19)(4 21 20)(5 22 16)(6 29 15)(7 30 11)(8 26 12)(9 27 13)(10 28 14)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 22)(12 23)(13 24)(14 25)(15 21)(16 30)(17 26)(18 27)(19 28)(20 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), (1,17,23)(2,18,24)(3,19,25)(4,20,21)(5,16,22)(6,29,15)(7,30,11)(8,26,12)(9,27,13)(10,28,14), (1,8)(2,9)(3,10)(4,6)(5,7)(11,16)(12,17)(13,18)(14,19)(15,20)(21,29)(22,30)(23,26)(24,27)(25,28), (1,23,17)(2,24,18)(3,25,19)(4,21,20)(5,22,16)(6,29,15)(7,30,11)(8,26,12)(9,27,13)(10,28,14), (1,8)(2,9)(3,10)(4,6)(5,7)(11,22)(12,23)(13,24)(14,25)(15,21)(16,30)(17,26)(18,27)(19,28)(20,29)>;

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

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

G:=TransitiveGroup(30,41);

C5×S32 is a maximal subgroup of   S32⋊D5

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + image C1 C2 C2 C5 C10 C10 S3 D6 C5×S3 S3×C10 S32 C5×S32 kernel C5×S32 S3×C15 C5×C3⋊S3 S32 C3×S3 C3⋊S3 C5×S3 C15 S3 C3 C5 C1 # reps 1 2 1 4 8 4 2 2 8 8 1 4

Matrix representation of C5×S32 in GL4(𝔽31) generated by

 8 0 0 0 0 8 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 30 1 0 0 30 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 30 1 0 0 30 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(31))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,30,30,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[30,30,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

C5×S32 in GAP, Magma, Sage, TeX

C_5\times S_3^2
% in TeX

G:=Group("C5xS3^2");
// GroupNames label

G:=SmallGroup(180,28);
// by ID

G=gap.SmallGroup(180,28);
# by ID

G:=PCGroup([5,-2,-2,-5,-3,-3,408,3004]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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