D9xC3:S3 - GroupNames

G = D₉×C₃⋊S₃ order 324 = 2²·3⁴

Direct product of D₉ and C₃⋊S₃

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

Aliases: D₉×C₃⋊S₃, C₃²⋊₇D₁₈, C₃³.₈D₆, (C₃×D₉)⋊S₃, C₃⋊₁(S₃×D₉), (C₃×C₉)⋊₁₃D₆, C₃².₈S₃², (C₃²×D₉)⋊₂C₂, (C₃²×C₉)⋊₃C₂², C₃²⋊₄D₉⋊₁C₂, C₉⋊₁(C₂×C₃⋊S₃), (C₉×C₃⋊S₃)⋊₂C₂, C₃.₂(S₃×C₃⋊S₃), (C₃×C₃⋊S₃).₄S₃, SmallGroup(324,119)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²×C₉ — D₉×C₃⋊S₃

Chief series

C₁ — C₃ — C₉ — C₃×C₉ — C₃²×C₉ — C₃²×D₉ — D₉×C₃⋊S₃

Lower central

C₃²×C₉ — D₉×C₃⋊S₃

Upper central

C₁

Generators and relations for D₉×C₃⋊S₃
G = < a,b,c,d,e | a⁹=b²=c³=d³=e²=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 1150 in 122 conjugacy classes, 29 normal (13 characteristic)
C₁, C₂, C₃, C₃, C₃, C₂², S₃, C₆, C₉, C₉, C₃², C₃², C₃², D₆, D₉, D₉, C₁₈, C₃×S₃, C₃⋊S₃, C₃⋊S₃, C₃×C₆, C₃×C₉, C₃×C₉, C₃³, D₁₈, S₃², C₂×C₃⋊S₃, C₃×D₉, S₃×C₉, C₉⋊S₃, S₃×C₃², C₃×C₃⋊S₃, C₃³⋊C₂, C₃²×C₉, S₃×D₉, S₃×C₃⋊S₃, C₃²×D₉, C₉×C₃⋊S₃, C₃²⋊₄D₉, D₉×C₃⋊S₃
Quotients: C₁, C₂, C₂², S₃, D₆, D₉, C₃⋊S₃, D₁₈, S₃², C₂×C₃⋊S₃, S₃×D₉, S₃×C₃⋊S₃, D₉×C₃⋊S₃

Smallest permutation representation of D₉×C₃⋊S₃
►On 54 points

Generators in S₅₄

(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)

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

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

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

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	3A	···	3E	3F	3G	3H	3I	6A	···	6E	9A	9B	9C	9D	···	9O	18A	18B	18C
order	1	2	2	2	3	···	3	3	3	3	3	6	···	6	9	9	9	9	···	9	18	18	18
size	1	9	9	81	2	···	2	4	4	4	4	18	···	18	2	2	2	4	···	4	18	18	18

36 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

D₉

D₁₈

S₃²

S₃×D₉

kernel

D₉×C₃⋊S₃

C₃²×D₉

C₉×C₃⋊S₃

C₃²⋊₄D₉

C₃×D₉

C₃×C₃⋊S₃

C₃×C₉

C₃³

C₃⋊S₃

C₃²

C₃

# reps

Matrix representation of D₉×C₃⋊S₃ ►in GL₆(𝔽₁₉)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	7	7
0	0	0	0	10	2

18	0	0	0	0	0
0	18	0	0	0	0
0	0	18	0	0	0
0	0	0	18	0	0
0	0	0	0	2	6
0	0	0	0	9	17

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

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

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

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,10,0,0,0,0,7,2],[18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,2,9,0,0,0,0,6,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,18,0,0,0,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,18,0,0,0,0,18,0,0,0,0,0,0,0,18,1,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18] >;

D₉×C₃⋊S₃ in GAP, Magma, Sage, TeX

D_9\times C_3\rtimes S_3

% in TeX

G:=Group("D9xC3:S3");

// GroupNames label

G:=SmallGroup(324,119);

// by ID

G=gap.SmallGroup(324,119);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,80,297,5404,208,7781]);

// Polycyclic

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

// generators/relations

G = D9×C3⋊S3 order 324 = 22·34

Direct product of D9 and C3⋊S3

G = D₉×C₃⋊S₃ order 324 = 2²·3⁴

Direct product of D₉ and C₃⋊S₃