C3^3:6D9 - GroupNames

G = C₃³⋊₆D₉ order 486 = 2·3⁵

6^th semidirect product of C₃³ and D₉ acting via D₉/C₃=S₃

Aliases: C₃³⋊₆D₉, C₃⁴.₈S₃, C₃²⋊C₉⋊₁₅S₃, C₃²⋊₄(C₉⋊S₃), (C₃²×C₉)⋊₁₇S₃, C₃⋊(C₃²⋊₂D₉), C₃³.₃₆(C₃⋊S₃), C₃.₁(He₃⋊₅S₃), C₃.₃(C₃²⋊₄D₉), C₃².₂₆(He₃⋊C₂), C₃².₁₀(C₃³⋊C₂), (C₃×C₉)⋊₄(C₃⋊S₃), (C₃×C₃²⋊C₉)⋊₇C₂, SmallGroup(486,181)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₃×C₃²⋊C₉ — C₃³⋊₆D₉

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃⁴ — C₃×C₃²⋊C₉ — C₃³⋊₆D₉

Lower central

C₃×C₃²⋊C₉ — C₃³⋊₆D₉

Upper central

C₁ — C₃

Generators and relations for C₃³⋊₆D₉
G = < a,b,c,d,e | a³=b³=c³=d⁹=e²=1, ab=ba, ac=ca, ad=da, eae=a^-1, bc=cb, dbd^-1=bc^-1, ebe=b^-1c, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 1600 in 207 conjugacy classes, 57 normal (9 characteristic)
C₁, C₂, C₃, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₃², D₉, C₃×S₃, C₃⋊S₃, C₃×C₉, C₃×C₉, C₃³, C₃³, C₃³, C₃×D₉, C₉⋊S₃, C₃×C₃⋊S₃, C₃³⋊C₂, C₃²⋊C₉, C₃²×C₉, C₃⁴, C₃²⋊₂D₉, C₃×C₉⋊S₃, C₃×C₃³⋊C₂, C₃×C₃²⋊C₉, C₃³⋊₆D₉
Quotients: C₁, C₂, S₃, D₉, C₃⋊S₃, C₉⋊S₃, He₃⋊C₂, C₃³⋊C₂, C₃²⋊₂D₉, C₃²⋊₄D₉, He₃⋊₅S₃, C₃³⋊₆D₉

Smallest permutation representation of C₃³⋊₆D₉
►On 54 points

Generators in S₅₄

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

(2 38 30)(3 31 39)(5 41 33)(6 34 42)(8 44 36)(9 28 45)(10 51 24)(11 25 52)(13 54 27)(14 19 46)(16 48 21)(17 22 49)

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

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

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

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

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

54 conjugacy classes

class	1	2	3A	3B	3C	···	3N	3O	···	3W	6A	6B	9A	···	9AA
order	1	2	3	3	3	···	3	3	···	3	6	6	9	···	9
size	1	81	1	1	2	···	2	6	···	6	81	81	6	···	6

54 irreducible representations

dim	1	1	2	2	2	2	3	6	6
type	+	+	+	+	+	+
image	C₁	C₂	S₃	S₃	S₃	D₉	He₃⋊C₂	C₃²⋊₂D₉	He₃⋊₅S₃
kernel	C₃³⋊₆D₉	C₃×C₃²⋊C₉	C₃²⋊C₉	C₃²×C₉	C₃⁴	C₃³	C₃²	C₃	C₃
# reps	1	1	9	3	1	27	4	6	2

Matrix representation of C₃³⋊₆D₉ ►in GL₇(𝔽₁₉)

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

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

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	11	0	0
0	0	0	0	0	11	0
0	0	0	0	0	0	11

9	0	0	0	0	0	0
0	17	0	0	0	0	0
0	0	17	0	0	0	0
0	0	0	9	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	11	0
0	0	0	0	0	0	7

0	16	0	0	0	0	0
6	0	0	0	0	0	0
0	0	0	4	0	0	0
0	0	5	0	0	0	0
0	0	0	0	18	0	0
0	0	0	0	0	0	12
0	0	0	0	0	8	0

G:=sub<GL(7,GF(19))| [7,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11],[9,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,7],[0,6,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,4,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,8,0,0,0,0,0,12,0] >;

C₃³⋊₆D₉ in GAP, Magma, Sage, TeX

C_3^3\rtimes_6D_9

% in TeX

G:=Group("C3^3:6D9");

// GroupNames label

G:=SmallGroup(486,181);

// by ID

G=gap.SmallGroup(486,181);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,697,655,218,867,2169,3244]);

// Polycyclic

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

// generators/relations

G = C33⋊6D9 order 486 = 2·35

6th semidirect product of C33 and D9 acting via D9/C3=S3

G = C₃³⋊₆D₉ order 486 = 2·3⁵

6^th semidirect product of C₃³ and D₉ acting via D₉/C₃=S₃