Copied to
clipboard

G = S32×C32order 324 = 22·34

Direct product of C32, S3 and S3

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

Aliases: S32×C32, C3318D6, C341C22, C322C62, C336(C2×C6), C329(S3×C6), (S3×C33)⋊1C2, (S3×C32)⋊3C6, C31(S3×C3×C6), (C3×S3)⋊(C3×C6), (C3×C3⋊S3)⋊3C6, C3⋊S32(C3×C6), (C32×C3⋊S3)⋊1C2, SmallGroup(324,165)

Series: Derived Chief Lower central Upper central

C1C32 — S32×C32
C1C3C32C33C34S3×C33 — S32×C32
C32 — S32×C32
C1C32

Generators and relations for S32×C32
 G = < a,b,c,d,e,f | a3=b3=c3=d2=e3=f2=1, ab=ba, 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: 664 in 224 conjugacy classes, 60 normal (8 characteristic)
C1, C2 [×3], C3 [×6], C3 [×13], C22, S3 [×2], S3 [×3], C6 [×22], C32 [×2], C32 [×8], C32 [×40], D6 [×2], C2×C6 [×4], C3×S3 [×10], C3×S3 [×20], C3⋊S3, C3×C6 [×19], C33 [×6], C33 [×13], S32, S3×C6 [×8], C62, S3×C32 [×10], S3×C32 [×11], C3×C3⋊S3 [×4], C32×C6 [×2], C34, C3×S32 [×4], S3×C3×C6 [×2], S3×C33 [×2], C32×C3⋊S3, S32×C32
Quotients: C1, C2 [×3], C3 [×4], C22, S3 [×2], C6 [×12], C32, D6 [×2], C2×C6 [×4], C3×S3 [×8], C3×C6 [×3], S32, S3×C6 [×8], C62, S3×C32 [×2], C3×S32 [×4], S3×C3×C6 [×2], S32×C32

Smallest permutation representation of S32×C32
On 36 points
Generators in S36
(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)(31 32 33)(34 35 36)
(1 15 10)(2 13 11)(3 14 12)(4 7 36)(5 8 34)(6 9 35)(16 19 22)(17 20 23)(18 21 24)(25 28 31)(26 29 32)(27 30 33)
(1 10 15)(2 11 13)(3 12 14)(4 7 36)(5 8 34)(6 9 35)(16 19 22)(17 20 23)(18 21 24)(25 31 28)(26 32 29)(27 33 30)
(1 21)(2 19)(3 20)(4 30)(5 28)(6 29)(7 33)(8 31)(9 32)(10 18)(11 16)(12 17)(13 22)(14 23)(15 24)(25 34)(26 35)(27 36)
(1 15 10)(2 13 11)(3 14 12)(4 36 7)(5 34 8)(6 35 9)(16 19 22)(17 20 23)(18 21 24)(25 31 28)(26 32 29)(27 33 30)
(1 30)(2 28)(3 29)(4 21)(5 19)(6 20)(7 24)(8 22)(9 23)(10 27)(11 25)(12 26)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)

G:=sub<Sym(36)| (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)(31,32,33)(34,35,36), (1,15,10)(2,13,11)(3,14,12)(4,7,36)(5,8,34)(6,9,35)(16,19,22)(17,20,23)(18,21,24)(25,28,31)(26,29,32)(27,30,33), (1,10,15)(2,11,13)(3,12,14)(4,7,36)(5,8,34)(6,9,35)(16,19,22)(17,20,23)(18,21,24)(25,31,28)(26,32,29)(27,33,30), (1,21)(2,19)(3,20)(4,30)(5,28)(6,29)(7,33)(8,31)(9,32)(10,18)(11,16)(12,17)(13,22)(14,23)(15,24)(25,34)(26,35)(27,36), (1,15,10)(2,13,11)(3,14,12)(4,36,7)(5,34,8)(6,35,9)(16,19,22)(17,20,23)(18,21,24)(25,31,28)(26,32,29)(27,33,30), (1,30)(2,28)(3,29)(4,21)(5,19)(6,20)(7,24)(8,22)(9,23)(10,27)(11,25)(12,26)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)>;

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)(31,32,33)(34,35,36), (1,15,10)(2,13,11)(3,14,12)(4,7,36)(5,8,34)(6,9,35)(16,19,22)(17,20,23)(18,21,24)(25,28,31)(26,29,32)(27,30,33), (1,10,15)(2,11,13)(3,12,14)(4,7,36)(5,8,34)(6,9,35)(16,19,22)(17,20,23)(18,21,24)(25,31,28)(26,32,29)(27,33,30), (1,21)(2,19)(3,20)(4,30)(5,28)(6,29)(7,33)(8,31)(9,32)(10,18)(11,16)(12,17)(13,22)(14,23)(15,24)(25,34)(26,35)(27,36), (1,15,10)(2,13,11)(3,14,12)(4,36,7)(5,34,8)(6,35,9)(16,19,22)(17,20,23)(18,21,24)(25,31,28)(26,32,29)(27,33,30), (1,30)(2,28)(3,29)(4,21)(5,19)(6,20)(7,24)(8,22)(9,23)(10,27)(11,25)(12,26)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36) );

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),(31,32,33),(34,35,36)], [(1,15,10),(2,13,11),(3,14,12),(4,7,36),(5,8,34),(6,9,35),(16,19,22),(17,20,23),(18,21,24),(25,28,31),(26,29,32),(27,30,33)], [(1,10,15),(2,11,13),(3,12,14),(4,7,36),(5,8,34),(6,9,35),(16,19,22),(17,20,23),(18,21,24),(25,31,28),(26,32,29),(27,33,30)], [(1,21),(2,19),(3,20),(4,30),(5,28),(6,29),(7,33),(8,31),(9,32),(10,18),(11,16),(12,17),(13,22),(14,23),(15,24),(25,34),(26,35),(27,36)], [(1,15,10),(2,13,11),(3,14,12),(4,36,7),(5,34,8),(6,35,9),(16,19,22),(17,20,23),(18,21,24),(25,31,28),(26,32,29),(27,33,30)], [(1,30),(2,28),(3,29),(4,21),(5,19),(6,20),(7,24),(8,22),(9,23),(10,27),(11,25),(12,26),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)])

81 conjugacy classes

class 1 2A2B2C3A···3H3I···3Z3AA···3AI6A···6P6Q···6AH6AI···6AP
order12223···33···33···36···66···66···6
size13391···12···24···43···36···69···9

81 irreducible representations

dim111111222244
type++++++
imageC1C2C2C3C6C6S3D6C3×S3S3×C6S32C3×S32
kernelS32×C32S3×C33C32×C3⋊S3C3×S32S3×C32C3×C3⋊S3S3×C32C33C3×S3C32C32C3
# reps121816822161618

Matrix representation of S32×C32 in GL4(𝔽7) generated by

1000
0100
0040
0004
,
4000
0400
0010
0001
,
1000
0100
0061
0060
,
1000
0100
0006
0060
,
0100
6600
0010
0001
,
1000
6600
0060
0006
G:=sub<GL(4,GF(7))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,6,6,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,6,0,0,6,0],[0,6,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[1,6,0,0,0,6,0,0,0,0,6,0,0,0,0,6] >;

S32×C32 in GAP, Magma, Sage, TeX

S_3^2\times C_3^2
% in TeX

G:=Group("S3^2xC3^2");
// GroupNames label

G:=SmallGroup(324,165);
// by ID

G=gap.SmallGroup(324,165);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,1090,7781]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^2=e^3=f^2=1,a*b=b*a,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

׿
×
𝔽