C3xC3^2:2D9 - GroupNames

G = C₃×C₃²⋊₂D₉ order 486 = 2·3⁵

Direct product of C₃ and C₃²⋊₂D₉

direct product, non-abelian, supersoluble, monomial

Aliases: C₃×C₃²⋊₂D₉, C₃³⋊₄D₉, C₃⁴.₅S₃, C₃²⋊₃(C₃×D₉), C₃²⋊C₉⋊₂₀C₆, (C₃²×C₉)⋊₁₂S₃, C₃³.₃₈(C₃×S₃), C₃³.₄₈(C₃⋊S₃), C₃².₂₀(C₉⋊S₃), C₃².₂₅(He₃⋊C₂), (C₃×C₉)⋊₁₂(C₃×S₃), C₃.₁₀(C₃×C₉⋊S₃), (C₃×C₃²⋊C₉)⋊₅C₂, C₃².₂₆(C₃×C₃⋊S₃), C₃.₁(C₃×He₃⋊C₂), SmallGroup(486,135)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₃²⋊C₉ — C₃×C₃²⋊₂D₉

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²⋊C₉ — C₃×C₃²⋊C₉ — C₃×C₃²⋊₂D₉

Lower central

C₃²⋊C₉ — C₃×C₃²⋊₂D₉

Upper central

C₁ — C₃²

Generators and relations for C₃×C₃²⋊₂D₉
G = < a,b,c,d,e | a³=b³=c³=d⁹=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=bc^-1, ebe=b^-1c, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 740 in 156 conjugacy classes, 30 normal (12 characteristic)
C₁, C₂, C₃, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₃², D₉, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₉, C₃×C₉, C₃³, C₃³, C₃³, C₃×D₉, S₃×C₃², C₃×C₃⋊S₃, C₃²⋊C₉, C₃²⋊C₉, C₃²×C₉, C₃⁴, C₃²⋊₂D₉, C₃²×D₉, C₃²×C₃⋊S₃, C₃×C₃²⋊C₉, C₃×C₃²⋊₂D₉
Quotients: C₁, C₂, C₃, S₃, C₆, D₉, C₃×S₃, C₃⋊S₃, C₃×D₉, C₉⋊S₃, He₃⋊C₂, C₃×C₃⋊S₃, C₃²⋊₂D₉, C₃×C₉⋊S₃, C₃×He₃⋊C₂, C₃×C₃²⋊₂D₉

Smallest permutation representation of C₃×C₃²⋊₂D₉
►On 54 points

Generators in S₅₄

(1 4 7)(2 5 8)(3 6 9)(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 52 49)(47 53 50)(48 54 51)

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

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

(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 49)(38 48)(39 47)(40 46)(41 54)(42 53)(43 52)(44 51)(45 50)

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

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

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

63 conjugacy classes

class	1	2	3A	···	3H	3I	···	3Q	3R	···	3Z	6A	···	6H	9A	···	9AA
order	1	2	3	···	3	3	···	3	3	···	3	6	···	6	9	···	9
size	1	27	1	···	1	2	···	2	6	···	6	27	···	27	6	···	6

63 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	3	6
type	+	+			+	+		+
image	C₁	C₂	C₃	C₆	S₃	S₃	C₃×S₃	D₉	C₃×S₃	C₃×D₉	He₃⋊C₂	C₃²⋊₂D₉
kernel	C₃×C₃²⋊₂D₉	C₃×C₃²⋊C₉	C₃²⋊₂D₉	C₃²⋊C₉	C₃²×C₉	C₃⁴	C₃×C₉	C₃³	C₃³	C₃²	C₃²	C₃
# reps	1	1	2	2	3	1	6	9	2	18	12	6

Matrix representation of C₃×C₃²⋊₂D₉ ►in GL₅(𝔽₁₉)

7	0	0	0	0
0	7	0	0	0
0	0	11	0	0
0	0	0	11	0
0	0	0	0	11

7	0	0	0	0
12	11	0	0	0
0	0	1	0	0
0	0	0	11	0
0	0	13	14	7

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

9	0	0	0	0
5	17	0	0	0
0	0	0	1	0
0	0	9	3	10
0	0	5	2	16

3	1	0	0	0
11	16	0	0	0
0	0	0	18	0
0	0	18	0	0
0	0	7	12	18

G:=sub<GL(5,GF(19))| [7,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[7,12,0,0,0,0,11,0,0,0,0,0,1,0,13,0,0,0,11,14,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[9,5,0,0,0,0,17,0,0,0,0,0,0,9,5,0,0,1,3,2,0,0,0,10,16],[3,11,0,0,0,1,16,0,0,0,0,0,0,18,7,0,0,18,0,12,0,0,0,0,18] >;

C₃×C₃²⋊₂D₉ in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_2D_9

% in TeX

G:=Group("C3xC3^2:2D9");

// GroupNames label

G:=SmallGroup(486,135);

// by ID

G=gap.SmallGroup(486,135);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,338,867,735,3244]);

// 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,a*e=e*a,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 = C3×C32⋊2D9 order 486 = 2·35

Direct product of C3 and C32⋊2D9

G = C₃×C₃²⋊₂D₉ order 486 = 2·3⁵

Direct product of C₃ and C₃²⋊₂D₉