Copied to
clipboard

## G = S32×C9order 324 = 22·34

### Direct product of C9, S3 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S32×C9
 Chief series C1 — C3 — C32 — C33 — C32×C9 — S3×C3×C9 — S32×C9
 Lower central C32 — S32×C9
 Upper central C1 — C9

Generators and relations for S32×C9
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, C2×C6, C18, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, C33, C2×C18, S32, S3×C6, S3×C9, S3×C9, C3×C18, S3×C32, C3×C3⋊S3, C32×C9, S3×C18, C3×S32, S3×C3×C9, C9×C3⋊S3, S32×C9
Quotients: C1, C2, C3, C22, S3, C6, C9, D6, C2×C6, C18, C3×S3, C2×C18, S32, S3×C6, S3×C9, S3×C18, C3×S32, S32×C9

Smallest permutation representation of S32×C9
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 C3×S3 S3×C6 S3×C9 S3×C18 S32 C3×S32 S32×C9 kernel S32×C9 S3×C3×C9 C9×C3⋊S3 C3×S32 S3×C32 C3×C3⋊S3 S32 C3×S3 C3⋊S3 S3×C9 C3×C9 C3×S3 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 S32×C9 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] >;

S32×C9 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

׿
×
𝔽