S3^2xC3^2 - GroupNames

G = S₃²×C₃² order 324 = 2²·3⁴

Direct product of C₃², S₃ and S₃

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

Aliases: S₃²×C₃², C₃³⋊₁₈D₆, C₃⁴⋊₁C₂², C₃²⋊₂C₆², C₃³⋊₆(C₂×C₆), C₃²⋊₉(S₃×C₆), (S₃×C₃³)⋊₁C₂, (S₃×C₃²)⋊₃C₆, C₃⋊₁(S₃×C₃×C₆), (C₃×S₃)⋊(C₃×C₆), (C₃×C₃⋊S₃)⋊₃C₆, C₃⋊S₃⋊₂(C₃×C₆), (C₃²×C₃⋊S₃)⋊₁C₂, SmallGroup(324,165)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — S₃²×C₃²

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃⁴ — S₃×C₃³ — S₃²×C₃²

Lower central

C₃² — S₃²×C₃²

Upper central

C₁ — C₃²

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, C₃, C₃, C₂², S₃, S₃, C₆, C₃², C₃², C₃², D₆, C₂×C₆, C₃×S₃, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃³, C₃³, S₃², S₃×C₆, C₆², S₃×C₃², S₃×C₃², C₃×C₃⋊S₃, C₃²×C₆, C₃⁴, C₃×S₃², S₃×C₃×C₆, S₃×C₃³, C₃²×C₃⋊S₃, S₃²×C₃²
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₃², D₆, C₂×C₆, C₃×S₃, C₃×C₆, S₃², S₃×C₆, C₆², S₃×C₃², C₃×S₃², S₃×C₃×C₆, 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	2A	2B	2C	3A	···	3H	3I	···	3Z	3AA	···	3AI	6A	···	6P	6Q	···	6AH	6AI	···	6AP
order	1	2	2	2	3	···	3	3	···	3	3	···	3	6	···	6	6	···	6	6	···	6
size	1	3	3	9	1	···	1	2	···	2	4	···	4	3	···	3	6	···	6	9	···	9

81 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+				+	+			+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₆	C₃×S₃	S₃×C₆	S₃²	C₃×S₃²
kernel	S₃²×C₃²	S₃×C₃³	C₃²×C₃⋊S₃	C₃×S₃²	S₃×C₃²	C₃×C₃⋊S₃	S₃×C₃²	C₃³	C₃×S₃	C₃²	C₃²	C₃
# reps	1	2	1	8	16	8	2	2	16	16	1	8

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

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	1
0	0	6	0

1	0	0	0
0	1	0	0
0	0	0	6
0	0	6	0

0	1	0	0
6	6	0	0
0	0	1	0
0	0	0	1

1	0	0	0
6	6	0	0
0	0	6	0
0	0	0	6

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

G = S32×C32 order 324 = 22·34

Direct product of C32, S3 and S3

G = S₃²×C₃² order 324 = 2²·3⁴

Direct product of C₃², S₃ and S₃