Copied to
clipboard

## G = S3×C32⋊C9order 486 = 2·35

### Direct product of S3 and C32⋊C9

Series: Derived Chief Lower central Upper central

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

Generators and relations for S3×C32⋊C9
G = < a,b,c,d,e | a3=b2=c3=d3=e9=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, de=ed >

Subgroups: 452 in 141 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C32, C18, C3×S3, C3×S3, C3×S3, C3×C6, C3×C9, C3×C9, C33, C33, C33, S3×C9, C3×C18, S3×C32, S3×C32, S3×C32, C32×C6, C32⋊C9, C32⋊C9, C32×C9, C34, C2×C32⋊C9, S3×C3×C9, S3×C33, C3×C32⋊C9, S3×C32⋊C9
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, He3, 3- 1+2, S3×C9, C3×C18, C2×He3, C2×3- 1+2, S3×C32, C32⋊C9, C2×C32⋊C9, S3×C3×C9, S3×He3, S3×3- 1+2, S3×C32⋊C9

Smallest permutation representation of S3×C32⋊C9
On 54 points
Generators in S54
(1 41 29)(2 42 30)(3 43 31)(4 44 32)(5 45 33)(6 37 34)(7 38 35)(8 39 36)(9 40 28)(10 23 47)(11 24 48)(12 25 49)(13 26 50)(14 27 51)(15 19 52)(16 20 53)(17 21 54)(18 22 46)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 45)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(28 51)(29 52)(30 53)(31 54)(32 46)(33 47)(34 48)(35 49)(36 50)
(1 7 4)(2 30 42)(3 37 28)(5 33 45)(6 40 31)(8 36 39)(9 43 34)(10 23 47)(11 51 21)(12 18 15)(13 26 50)(14 54 24)(16 20 53)(17 48 27)(19 25 22)(29 35 32)(38 44 41)(46 52 49)
(1 38 32)(2 39 33)(3 40 34)(4 41 35)(5 42 36)(6 43 28)(7 44 29)(8 45 30)(9 37 31)(10 53 26)(11 54 27)(12 46 19)(13 47 20)(14 48 21)(15 49 22)(16 50 23)(17 51 24)(18 52 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 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)

G:=sub<Sym(54)| (1,41,29)(2,42,30)(3,43,31)(4,44,32)(5,45,33)(6,37,34)(7,38,35)(8,39,36)(9,40,28)(10,23,47)(11,24,48)(12,25,49)(13,26,50)(14,27,51)(15,19,52)(16,20,53)(17,21,54)(18,22,46), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,45)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(28,51)(29,52)(30,53)(31,54)(32,46)(33,47)(34,48)(35,49)(36,50), (1,7,4)(2,30,42)(3,37,28)(5,33,45)(6,40,31)(8,36,39)(9,43,34)(10,23,47)(11,51,21)(12,18,15)(13,26,50)(14,54,24)(16,20,53)(17,48,27)(19,25,22)(29,35,32)(38,44,41)(46,52,49), (1,38,32)(2,39,33)(3,40,34)(4,41,35)(5,42,36)(6,43,28)(7,44,29)(8,45,30)(9,37,31)(10,53,26)(11,54,27)(12,46,19)(13,47,20)(14,48,21)(15,49,22)(16,50,23)(17,51,24)(18,52,25), (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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)>;

G:=Group( (1,41,29)(2,42,30)(3,43,31)(4,44,32)(5,45,33)(6,37,34)(7,38,35)(8,39,36)(9,40,28)(10,23,47)(11,24,48)(12,25,49)(13,26,50)(14,27,51)(15,19,52)(16,20,53)(17,21,54)(18,22,46), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,45)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(28,51)(29,52)(30,53)(31,54)(32,46)(33,47)(34,48)(35,49)(36,50), (1,7,4)(2,30,42)(3,37,28)(5,33,45)(6,40,31)(8,36,39)(9,43,34)(10,23,47)(11,51,21)(12,18,15)(13,26,50)(14,54,24)(16,20,53)(17,48,27)(19,25,22)(29,35,32)(38,44,41)(46,52,49), (1,38,32)(2,39,33)(3,40,34)(4,41,35)(5,42,36)(6,43,28)(7,44,29)(8,45,30)(9,37,31)(10,53,26)(11,54,27)(12,46,19)(13,47,20)(14,48,21)(15,49,22)(16,50,23)(17,51,24)(18,52,25), (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)(28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54) );

G=PermutationGroup([[(1,41,29),(2,42,30),(3,43,31),(4,44,32),(5,45,33),(6,37,34),(7,38,35),(8,39,36),(9,40,28),(10,23,47),(11,24,48),(12,25,49),(13,26,50),(14,27,51),(15,19,52),(16,20,53),(17,21,54),(18,22,46)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,45),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(28,51),(29,52),(30,53),(31,54),(32,46),(33,47),(34,48),(35,49),(36,50)], [(1,7,4),(2,30,42),(3,37,28),(5,33,45),(6,40,31),(8,36,39),(9,43,34),(10,23,47),(11,51,21),(12,18,15),(13,26,50),(14,54,24),(16,20,53),(17,48,27),(19,25,22),(29,35,32),(38,44,41),(46,52,49)], [(1,38,32),(2,39,33),(3,40,34),(4,41,35),(5,42,36),(6,43,28),(7,44,29),(8,45,30),(9,37,31),(10,53,26),(11,54,27),(12,46,19),(13,47,20),(14,48,21),(15,49,22),(16,50,23),(17,51,24),(18,52,25)], [(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),(28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)]])

99 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 3R ··· 3W 3X ··· 3AC 6A ··· 6H 6I ··· 6N 9A ··· 9R 9S ··· 9AJ 18A ··· 18R order 1 2 3 ··· 3 3 ··· 3 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 ··· 1 2 ··· 2 3 ··· 3 6 ··· 6 3 ··· 3 9 ··· 9 3 ··· 3 6 ··· 6 9 ··· 9

99 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 6 6 type + + + image C1 C2 C3 C3 C6 C6 C9 C18 S3 C3×S3 C3×S3 S3×C9 He3 3- 1+2 C2×He3 C2×3- 1+2 S3×He3 S3×3- 1+2 kernel S3×C32⋊C9 C3×C32⋊C9 S3×C3×C9 S3×C33 C32×C9 C34 S3×C32 C33 C32⋊C9 C3×C9 C33 C32 C3×S3 C3×S3 C32 C32 C3 C3 # reps 1 1 6 2 6 2 18 18 1 6 2 18 2 4 2 4 2 4

Matrix representation of S3×C32⋊C9 in GL5(𝔽19)

 7 0 0 0 0 11 11 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 9 0 0 0 0 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 12 0 0 0 11 1 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 4 0 0 0 0 0 4 0 0 0 0 0 8 7 11 0 0 8 0 0 0 0 15 0 11

G:=sub<GL(5,GF(19))| [7,11,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,9,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,12,1,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[4,0,0,0,0,0,4,0,0,0,0,0,8,8,15,0,0,7,0,0,0,0,11,0,11] >;

S3×C32⋊C9 in GAP, Magma, Sage, TeX

S_3\times C_3^2\rtimes C_9
% in TeX

G:=Group("S3xC3^2:C9");
// GroupNames label

G:=SmallGroup(486,95);
// by ID

G=gap.SmallGroup(486,95);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,68,11669]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^3=e^9=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,d*e=e*d>;
// generators/relations

׿
×
𝔽