Copied to
clipboard

G = C5xS3xD5order 300 = 22·3·52

Direct product of C5, S3 and D5

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

Aliases: C5xS3xD5, D15:C10, C52:5D6, C15:5D10, (C5xS3):C10, C15:(C2xC10), (C3xD5):C10, C3:1(D5xC10), C5:1(S3xC10), (C5xD15):1C2, (D5xC15):3C2, (C5xC15):1C22, (S3xC52):1C2, SmallGroup(300,37)

Series: Derived Chief Lower central Upper central

C1C15 — C5xS3xD5
C1C5C15C5xC15D5xC15 — C5xS3xD5
C15 — C5xS3xD5
C1C5

Generators and relations for C5xS3xD5
 G = < a,b,c,d,e | a5=b3=c2=d5=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: 168 in 48 conjugacy classes, 20 normal (all characteristic)
Quotients: C1, C2, C22, C5, S3, D5, C10, D6, D10, C2xC10, C5xS3, C5xD5, S3xD5, S3xC10, D5xC10, C5xS3xD5
3C2
5C2
15C2
2C5
2C5
15C22
5S3
5C6
3D5
3C10
3C10
5C10
6C10
6C10
15C10
2C15
2C15
5D6
3D10
15C2xC10
2C5xS3
2C5xS3
5C5xS3
5C30
3C5xD5
3C5xC10
5S3xC10
3D5xC10

Permutation representations of C5xS3xD5
On 30 points - transitive group 30T75
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 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 5 4 3 2)(6 10 9 8 7)(11 15 14 13 12)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 20)(2 16)(3 17)(4 18)(5 19)(6 25)(7 21)(8 22)(9 23)(10 24)(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), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30), (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,12)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,20)(2,16)(3,17)(4,18)(5,19)(6,25)(7,21)(8,22)(9,23)(10,24)(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), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30), (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,12)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,20)(2,16)(3,17)(4,18)(5,19)(6,25)(7,21)(8,22)(9,23)(10,24)(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)], [(1,6,11),(2,7,12),(3,8,13),(4,9,14),(5,10,15),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(6,11),(7,12),(8,13),(9,14),(10,15),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,5,4,3,2),(6,10,9,8,7),(11,15,14,13,12),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,20),(2,16),(3,17),(4,18),(5,19),(6,25),(7,21),(8,22),(9,23),(10,24),(11,30),(12,26),(13,27),(14,28),(15,29)]])

G:=TransitiveGroup(30,75);

60 conjugacy classes

class 1 2A2B2C 3 5A5B5C5D5E···5N 6 10A10B10C10D10E10F10G10H10I···10R10S10T10U10V15A15B15C15D15E···15N30A30B30C30D
order1222355555···56101010101010101010···10101010101515151515···1530303030
size13515211112···210333355556···61515151522224···410101010

60 irreducible representations

dim111111112222222244
type+++++++++
imageC1C2C2C2C5C10C10C10S3D5D6D10C5xS3C5xD5S3xC10D5xC10S3xD5C5xS3xD5
kernelC5xS3xD5D5xC15S3xC52C5xD15S3xD5C5xS3C3xD5D15C5xD5C5xS3C52C15D5S3C5C3C5C1
# reps111144441212484828

Matrix representation of C5xS3xD5 in GL4(F31) generated by

2000
0200
0040
0004
,
1000
0100
00030
00130
,
1000
0100
00130
00030
,
4000
0800
0010
0001
,
0800
4000
00300
00030
G:=sub<GL(4,GF(31))| [2,0,0,0,0,2,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,30,30],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,30,30],[4,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,4,0,0,8,0,0,0,0,0,30,0,0,0,0,30] >;

C5xS3xD5 in GAP, Magma, Sage, TeX

C_5\times S_3\times D_5
% in TeX

G:=Group("C5xS3xD5");
// GroupNames label

G:=SmallGroup(300,37);
// by ID

G=gap.SmallGroup(300,37);
# by ID

G:=PCGroup([5,-2,-2,-5,-3,-5,408,6004]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^5=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 C5xS3xD5 in TeX

׿
x
:
Z
F
o
wr
Q
<