C3^2:C9:C6 - GroupNames

G = C₃²:C₉:C₆ order 486 = 2·3⁵

1^st semidirect product of C₃²:C₉ and C₆ acting faithfully

Aliases: C₃.₂C₃wrS₃, C₃²:C₉:₁C₆, (C₃xHe₃).₁S₃, C₃³.₃(C₃xS₃), C₃²:₂D₉:₂C₃, C₃³.C₃²:₂C₂, C₃².₂₇(C₃²:C₆), C₃.₃(He₃.S₃), SmallGroup(486,6)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — C₃²:C₉ — C₃²:C₉:C₆

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²:C₉ — C₃³.C₃² — C₃²:C₉:C₆

Lower central

C₃²:C₉ — C₃²:C₉:C₆

Upper central

C₁ — C₃

Generators and relations for C₃²:C₉:C₆
G = < a,b,c,d | a³=b³=c⁹=d⁶=1, ab=ba, cac^-1=ab^-1, dad^-1=a^-1bc⁶, bc=cb, bd=db, dcd^-1=a^-1c⁵ >

Subgroups: 470 in 59 conjugacy classes, 10 normal (all characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², D₉, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₉, He₃, 3_-¹⁺², C₃³, C₃³, C₃xD₉, C₃²:C₆, S₃xC₃², C₃xC₃:S₃, C₃²:C₉, C₃xHe₃, C₃x3_-¹⁺², C₃²:₂D₉, C₃xC₃²:C₆, C₃³.C₃², C₃²:C₉:C₆
Quotients: C₁, C₂, C₃, S₃, C₆, C₃xS₃, C₃²:C₆, C₃wrS₃, He₃.S₃, C₃²:C₉:C₆

Permutation representations of C₃²:C₉:C₆
►On 18 points - transitive group 18T174

Generators in S₁₈

(1 7 4)(3 6 9)(11 17 14)(12 15 18)

(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)

(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)

(1 15 9 16 2 17)(3 13 5 14 4 12)(6 10 8 11 7 18)

G:=sub<Sym(18)| (1,7,4)(3,6,9)(11,17,14)(12,15,18), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,15,9,16,2,17)(3,13,5,14,4,12)(6,10,8,11,7,18)>;

G:=Group( (1,7,4)(3,6,9)(11,17,14)(12,15,18), (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,15,9,16,2,17)(3,13,5,14,4,12)(6,10,8,11,7,18) );

G=PermutationGroup([[(1,7,4),(3,6,9),(11,17,14),(12,15,18)], [(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,15,9,16,2,17),(3,13,5,14,4,12),(6,10,8,11,7,18)]])

G:=TransitiveGroup(18,174);

31 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	···	3K	3L	6A	···	6H	9A	···	9I
order	1	2	3	3	3	3	3	3	···	3	3	6	···	6	9	···	9
size	1	27	1	1	2	2	2	9	···	9	18	27	···	27	18	···	18

31 irreducible representations

dim	1	1	1	1	2	2	3	6	6	6
type	+	+			+			+	+
image	C₁	C₂	C₃	C₆	S₃	C₃xS₃	C₃wrS₃	C₃²:C₆	He₃.S₃	C₃²:C₉:C₆
kernel	C₃²:C₉:C₆	C₃³.C₃²	C₃²:₂D₉	C₃²:C₉	C₃xHe₃	C₃³	C₃	C₃²	C₃	C₁
# reps	1	1	2	2	1	2	12	1	3	6

Matrix representation of C₃²:C₉:C₆ ►in GL₆(F₁₉)

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

7	0	0	0	0	0
0	7	0	0	0	0
0	0	7	0	0	0
0	0	0	7	0	0
0	0	0	0	7	0
0	0	0	0	0	7

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

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

G:=sub<GL(6,GF(19))| [7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,0,1,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0,7,0,0,0,0,0,0,11,0,0,0,1,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0] >;

C₃²:C₉:C₆ in GAP, Magma, Sage, TeX

C_3^2\rtimes C_9\rtimes C_6

% in TeX

G:=Group("C3^2:C9:C6");

// GroupNames label

G:=SmallGroup(486,6);

// by ID

G=gap.SmallGroup(486,6);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,2162,224,176,4755,873,735,3244]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^9=d^6=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^-1*b*c^6,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*c^5>;

// generators/relations

G = C32:C9:C6 order 486 = 2·35

1st semidirect product of C32:C9 and C6 acting faithfully

G = C₃²:C₉:C₆ order 486 = 2·3⁵

1^st semidirect product of C₃²:C₉ and C₆ acting faithfully