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G = C5xS32order 180 = 22·32·5

Direct product of C5, S3 and S3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C5xS32, C15:5D6, C3:S3:C10, (C3xS3):C10, C32:(C2xC10), C3:1(S3xC10), (S3xC15):3C2, (C3xC15):5C22, (C5xC3:S3):3C2, SmallGroup(180,28)

Series: Derived Chief Lower central Upper central

C1C32 — C5xS32
C1C3C32C3xC15S3xC15 — C5xS32
C32 — C5xS32
C1C5

Generators and relations for C5xS32
 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 >

Subgroups: 120 in 44 conjugacy classes, 20 normal (8 characteristic)
Quotients: C1, C2, C22, C5, S3, C10, D6, C2xC10, C5xS3, S32, S3xC10, C5xS32
3C2
3C2
9C2
2C3
9C22
3S3
3C6
3S3
3C6
6S3
3C10
3C10
9C10
2C15
3D6
3D6
9C2xC10
3C30
3C5xS3
3C30
3C5xS3
6C5xS3
3S3xC10
3S3xC10

Permutation representations of C5xS32
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);

C5xS32 is a maximal subgroup of   S32:D5

45 conjugacy classes

class 1 2A2B2C3A3B3C5A5B5C5D6A6B10A···10H10I10J10K10L15A···15H15I15J15K15L30A···30H
order122233355556610···101010101015···151515151530···30
size13392241111663···399992···244446···6

45 irreducible representations

dim111111222244
type++++++
imageC1C2C2C5C10C10S3D6C5xS3S3xC10S32C5xS32
kernelC5xS32S3xC15C5xC3:S3S32C3xS3C3:S3C5xS3C15S3C3C5C1
# reps121484228814

Matrix representation of C5xS32 in GL4(F31) generated by

8000
0800
0080
0008
,
1000
0100
00301
00300
,
1000
0100
0001
0010
,
30100
30000
0010
0001
,
0100
1000
0010
0001
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] >;

C5xS32 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

Export

Subgroup lattice of C5xS32 in TeX

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