C3xC3wrS3 - GroupNames

G = C₃xC₃wrS₃ order 486 = 2·3⁵

Direct product of C₃ and C₃wrS₃

direct product, non-abelian, supersoluble, monomial

Aliases: C₃xC₃wrS₃, C₃⁴:₂S₃, C₃wrC₃:₇C₆, He₃:(C₃xC₆), (C₃xHe₃):₄C₆, C₃³:₇(C₃xS₃), He₃:C₂:₁C₃², C₃².₁(S₃xC₃²), C₃².₄₉(C₃²:C₆), (C₃xC₃wrC₃):₃C₂, (C₃xHe₃:C₂):₁C₃, C₃.₁₅(C₃xC₃²:C₆), SmallGroup(486,115)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — He₃ — C₃xC₃wrS₃

Chief series

C₁ — C₃ — C₃² — He₃ — C₃xHe₃ — C₃xC₃wrC₃ — C₃xC₃wrS₃

Lower central

He₃ — C₃xC₃wrS₃

Upper central

C₁ — C₃²

Generators and relations for C₃xC₃wrS₃
G = < a,b,c,d,e | a³=b³=c³=d³=e⁶=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=bc^-1, ebe^-1=b^-1c, cd=dc, ce=ec, ede^-1=b^-1d^-1 >

Subgroups: 738 in 165 conjugacy classes, 24 normal (12 characteristic)
C₁, C₂, C₃, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₃xS₃, C₃xC₆, C₃xC₉, He₃, He₃, 3_-¹⁺², C₃³, C₃³, C₃³, He₃:C₂, S₃xC₃², C₃²xC₆, C₃wrC₃, C₃wrC₃, C₃xHe₃, C₃x3_-¹⁺², C₃⁴, C₃wrS₃, C₃xHe₃:C₂, S₃xC₃³, C₃xC₃wrC₃, C₃xC₃wrS₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃xS₃, C₃xC₆, C₃²:C₆, S₃xC₃², C₃wrS₃, C₃xC₃²:C₆, C₃xC₃wrS₃

Permutation representations of C₃xC₃wrS₃
►On 27 points - transitive group 27T177

Generators in S₂₇

(1 7 6)(2 8 4)(3 9 5)(10 12 14)(11 13 15)(16 18 20)(17 19 21)(22 24 26)(23 25 27)

(1 5 8)(2 6 9)(3 4 7)(11 19 23)(13 21 25)(15 17 27)

(1 8 5)(2 9 6)(3 7 4)(10 18 22)(11 19 23)(12 20 24)(13 21 25)(14 16 26)(15 17 27)

(1 13 10)(2 19 16)(3 27 24)(4 17 20)(5 25 22)(6 11 14)(7 15 12)(8 21 18)(9 23 26)

(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)

G:=sub<Sym(27)| (1,7,6)(2,8,4)(3,9,5)(10,12,14)(11,13,15)(16,18,20)(17,19,21)(22,24,26)(23,25,27), (1,5,8)(2,6,9)(3,4,7)(11,19,23)(13,21,25)(15,17,27), (1,8,5)(2,9,6)(3,7,4)(10,18,22)(11,19,23)(12,20,24)(13,21,25)(14,16,26)(15,17,27), (1,13,10)(2,19,16)(3,27,24)(4,17,20)(5,25,22)(6,11,14)(7,15,12)(8,21,18)(9,23,26), (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)>;

G:=Group( (1,7,6)(2,8,4)(3,9,5)(10,12,14)(11,13,15)(16,18,20)(17,19,21)(22,24,26)(23,25,27), (1,5,8)(2,6,9)(3,4,7)(11,19,23)(13,21,25)(15,17,27), (1,8,5)(2,9,6)(3,7,4)(10,18,22)(11,19,23)(12,20,24)(13,21,25)(14,16,26)(15,17,27), (1,13,10)(2,19,16)(3,27,24)(4,17,20)(5,25,22)(6,11,14)(7,15,12)(8,21,18)(9,23,26), (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) );

G=PermutationGroup([[(1,7,6),(2,8,4),(3,9,5),(10,12,14),(11,13,15),(16,18,20),(17,19,21),(22,24,26),(23,25,27)], [(1,5,8),(2,6,9),(3,4,7),(11,19,23),(13,21,25),(15,17,27)], [(1,8,5),(2,9,6),(3,7,4),(10,18,22),(11,19,23),(12,20,24),(13,21,25),(14,16,26),(15,17,27)], [(1,13,10),(2,19,16),(3,27,24),(4,17,20),(5,25,22),(6,11,14),(7,15,12),(8,21,18),(9,23,26)], [(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)]])

G:=TransitiveGroup(27,177);

66 conjugacy classes

class	1	2	3A	···	3H	3I	···	3Z	3AA	3AB	3AC	3AD	3AE	3AF	6A	···	6Z	9A	···	9F
order	1	2	3	···	3	3	···	3	3	3	3	3	3	3	6	···	6	9	···	9
size	1	9	1	···	1	3	···	3	6	6	6	18	18	18	9	···	9	18	···	18

66 irreducible representations

dim	1	1	1	1	1	1	2	2	3	6	6
type	+	+					+			+
image	C₁	C₂	C₃	C₃	C₆	C₆	S₃	C₃xS₃	C₃wrS₃	C₃²:C₆	C₃xC₃²:C₆
kernel	C₃xC₃wrS₃	C₃xC₃wrC₃	C₃wrS₃	C₃xHe₃:C₂	C₃wrC₃	C₃xHe₃	C₃⁴	C₃³	C₃	C₃²	C₃
# reps	1	1	6	2	6	2	1	8	36	1	2

Matrix representation of C₃xC₃wrS₃ ►in GL₅(F₁₉)

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

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

0	1	0	0	0
18	18	0	0	0
0	0	0	11	0
0	0	0	0	1
0	0	7	0	0

0	7	0	0	0
7	0	0	0	0
0	0	7	0	0
0	0	0	0	1
0	0	0	1	0

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

C₃xC₃wrS₃ in GAP, Magma, Sage, TeX

C_3\times C_3\wr S_3

% in TeX

G:=Group("C3xC3wrS3");

// GroupNames label

G:=SmallGroup(486,115);

// by ID

G=gap.SmallGroup(486,115);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,867,873,8104,382]);

// Polycyclic

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

// generators/relations

G = C3xC3wrS3 order 486 = 2·35

Direct product of C3 and C3wrS3

G = C₃xC₃wrS₃ order 486 = 2·3⁵

Direct product of C₃ and C₃wrS₃