S3^2xC9 - GroupNames

G = S₃²xC₉ order 324 = 2²·3⁴

Direct product of C₉, S₃ and S₃

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

Aliases: S₃²xC₉, (C₃xS₃):C₁₈, C₃:S₃:₂C₁₈, C₃:₁(S₃xC₁₈), (C₃xC₉):₁₅D₆, C₃²:₂(C₂xC₁₈), C₃³.₃(C₂xC₆), (C₃²xC₉):₁C₂², (S₃xC₃²).₂C₆, C₃².₁₂(S₃xC₆), (C₃xS₃²).C₃, (S₃xC₃xC₉):₁C₂, C₃.₅(C₃xS₃²), (C₉xC₃:S₃):₁C₂, (C₃xC₃:S₃).₃C₆, (C₃xS₃).₄(C₃xS₃), SmallGroup(324,115)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — S₃²xC₉

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²xC₉ — S₃xC₃xC₉ — S₃²xC₉

Lower central

C₃² — S₃²xC₉

Upper central

C₁ — C₉

Generators and relations for S₃²xC₉
G = < a,b,c,d,e | a⁹=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: 244 in 86 conjugacy classes, 30 normal (12 characteristic)
C₁, C₂, C₃, C₃, C₃, C₂², S₃, S₃, C₆, C₉, C₉, C₃², C₃², C₃², D₆, C₂xC₆, C₁₈, C₃xS₃, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₉, C₃xC₉, C₃³, C₂xC₁₈, S₃², S₃xC₆, S₃xC₉, S₃xC₉, C₃xC₁₈, S₃xC₃², C₃xC₃:S₃, C₃²xC₉, S₃xC₁₈, C₃xS₃², S₃xC₃xC₉, C₉xC₃:S₃, S₃²xC₉
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₉, D₆, C₂xC₆, C₁₈, C₃xS₃, C₂xC₁₈, S₃², S₃xC₆, S₃xC₉, S₃xC₁₈, C₃xS₃², S₃²xC₉

Smallest permutation representation of S₃²xC₉
►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 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)

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

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

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

81 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3H	3I	3J	3K	6A	6B	6C	6D	6E	···	6J	6K	6L	9A	···	9F	9G	···	9R	9S	···	9X	18A	···	18L	18M	···	18X	18Y	···	18AD
order	1	2	2	2	3	3	3	···	3	3	3	3	6	6	6	6	6	···	6	6	6	9	···	9	9	···	9	9	···	9	18	···	18	18	···	18	18	···	18
size	1	3	3	9	1	1	2	···	2	4	4	4	3	3	3	3	6	···	6	9	9	1	···	1	2	···	2	4	···	4	3	···	3	6	···	6	9	···	9

81 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4
type	+	+	+							+	+					+
image	C₁	C₂	C₂	C₃	C₆	C₆	C₉	C₁₈	C₁₈	S₃	D₆	C₃xS₃	S₃xC₆	S₃xC₉	S₃xC₁₈	S₃²	C₃xS₃²	S₃²xC₉
kernel	S₃²xC₉	S₃xC₃xC₉	C₉xC₃:S₃	C₃xS₃²	S₃xC₃²	C₃xC₃:S₃	S₃²	C₃xS₃	C₃:S₃	S₃xC₉	C₃xC₉	C₃xS₃	C₃²	S₃	C₃	C₉	C₃	C₁
# reps	1	2	1	2	4	2	6	12	6	2	2	4	4	12	12	1	2	6

Matrix representation of S₃²xC₉ ►in GL₄(F₁₉) generated by

7	0	0	0
0	7	0	0
0	0	5	0
0	0	0	5

18	1	0	0
18	0	0	0
0	0	1	0
0	0	0	1

0	18	0	0
18	0	0	0
0	0	18	0
0	0	0	18

1	0	0	0
0	1	0	0
0	0	11	0
0	0	0	7

1	0	0	0
0	1	0	0
0	0	0	16
0	0	6	0

G:=sub<GL(4,GF(19))| [7,0,0,0,0,7,0,0,0,0,5,0,0,0,0,5],[18,18,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,18,0,0,18,0,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,11,0,0,0,0,7],[1,0,0,0,0,1,0,0,0,0,0,6,0,0,16,0] >;

S₃²xC₉ in GAP, Magma, Sage, TeX

S_3^2\times C_9

% in TeX

G:=Group("S3^2xC9");

// GroupNames label

G:=SmallGroup(324,115);

// by ID

G=gap.SmallGroup(324,115);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,68,1090,7781]);

// Polycyclic

G:=Group<a,b,c,d,e|a^9=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

G = S32xC9 order 324 = 22·34

Direct product of C9, S3 and S3

G = S₃²xC₉ order 324 = 2²·3⁴

Direct product of C₉, S₃ and S₃