Copied to
clipboard

G = S32xD5order 360 = 23·32·5

Direct product of S3, S3 and D5

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

Aliases: S32xD5, D15:1D6, (S3xD15):C2, (C3xC15):C23, C3:S3:1D10, (C3xD5):1D6, (C5xS3):1D6, C3:D15:C22, D15:S3:3C2, (C3xS3):1D10, (S3xC15):C22, (C3xD15):C22, C15:1(C22xS3), (C32xD5):C22, C32:1(C22xD5), C5:1(C2xS32), (C3xS3xD5):C2, (D5xC3:S3):C2, C3:1(C2xS3xD5), (C5xS32):1C2, (C5xC3:S3):C22, SmallGroup(360,137)

Series: Derived Chief Lower central Upper central

C1C3xC15 — S32xD5
C1C5C15C3xC15C32xD5C3xS3xD5 — S32xD5
C3xC15 — S32xD5
C1

Generators and relations for S32xD5
 G = < a,b,c,d,e,f | a3=b2=c3=d2=e5=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 996 in 138 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C3, C3, C22, C5, S3, S3, C6, C23, C32, D5, D5, C10, D6, C2xC6, C15, C15, C3xS3, C3xS3, C3:S3, C3:S3, C3xC6, D10, C2xC10, C22xS3, C5xS3, C5xS3, C3xD5, C3xD5, D15, D15, C30, S32, S32, S3xC6, C2xC3:S3, C22xD5, C3xC15, S3xD5, S3xD5, C6xD5, S3xC10, D30, C2xS32, C32xD5, S3xC15, C3xD15, C5xC3:S3, C3:D15, C2xS3xD5, C3xS3xD5, D5xC3:S3, C5xS32, S3xD15, D15:S3, S32xD5
Quotients: C1, C2, C22, S3, C23, D5, D6, D10, C22xS3, S32, C22xD5, S3xD5, C2xS32, C2xS3xD5, S32xD5

Permutation representations of S32xD5
On 30 points - transitive group 30T84
Generators in S30
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 26)(7 27)(8 28)(9 29)(10 30)(11 21)(12 22)(13 23)(14 24)(15 25)
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)

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

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

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

G:=TransitiveGroup(30,84);

36 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C5A5B6A6B6C6D6E6F6G10A10B10C10D10E10F15A15B15C15D15E15F30A30B30C30D
order1222222233355666666610101010101015151515151530303030
size13359151545224226610102030306666181844448812121212

36 irreducible representations

dim111111222222244448
type++++++++++++++++++
imageC1C2C2C2C2C2S3D5D6D6D6D10D10S32S3xD5C2xS32C2xS3xD5S32xD5
kernelS32xD5C3xS3xD5D5xC3:S3C5xS32S3xD15D15:S3S3xD5S32C5xS3C3xD5D15C3xS3C3:S3D5S3C5C3C1
# reps121121222224214142

Matrix representation of S32xD5 in GL6(F31)

100000
010000
001000
000100
0000301
0000300
,
100000
010000
0030000
0003000
000001
000010
,
100000
010000
0030100
0030000
000010
000001
,
3000000
0300000
0003000
0030000
000010
000001
,
1310000
17300000
001000
000100
000010
000001
,
110000
0300000
0030000
0003000
000010
000001

G:=sub<GL(6,GF(31))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,30,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,30,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,0,30,0,0,0,0,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,17,0,0,0,0,1,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,1,30,0,0,0,0,0,0,30,0,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

S32xD5 in GAP, Magma, Sage, TeX

S_3^2\times D_5
% in TeX

G:=Group("S3^2xD5");
// GroupNames label

G:=SmallGroup(360,137);
// by ID

G=gap.SmallGroup(360,137);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,111,730,10373]);
// Polycyclic

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

׿
x
:
Z
F
o
wr
Q
<