C3xC3^3:S3 - GroupNames

G = C₃×C₃³⋊S₃ order 486 = 2·3⁵

Direct product of C₃ and C₃³⋊S₃

direct product, non-abelian, supersoluble, monomial

Aliases: C₃×C₃³⋊S₃, C₃⁴⋊₄S₃, C₃≀C₃⋊₆C₆, (C₃×He₃)⋊₉S₃, He₃⋊₂(C₃×S₃), C₃³⋊₈(C₃×S₃), C₃³.₃₂(C₃⋊S₃), 3_-¹⁺²⋊₁(C₃×S₃), (C₃×3_-¹⁺²)⋊₅S₃, C₃².₄(He₃⋊C₂), (C₃×C₃≀C₃)⋊₁C₂, C₃².₁(C₃×C₃⋊S₃), C₃.₅(C₃×He₃⋊C₂), SmallGroup(486,165)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃≀C₃ — C₃×C₃³⋊S₃

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃≀C₃ — C₃×C₃≀C₃ — C₃×C₃³⋊S₃

Lower central

C₃≀C₃ — C₃×C₃³⋊S₃

Upper central

C₁ — C₃

Generators and relations for C₃×C₃³⋊S₃
G = < a,b,c,d,e,f | a³=b³=c³=d³=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe^-1=bc=cb, bd=db, fbf=b^-1, ece^-1=cd=dc, fcf=cd^-1, de=ed, fdf=d^-1, fef=e^-1 >

Subgroups: 848 in 150 conjugacy classes, 20 normal (14 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₃², D₉, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₉, He₃, He₃, 3_-¹⁺², 3_-¹⁺², C₃³, C₃³, C₃×D₉, C₃²⋊C₆, C₉⋊C₆, S₃×C₃², C₃×C₃⋊S₃, C₃≀C₃, C₃≀C₃, C₃×He₃, C₃×3_-¹⁺², C₃⁴, C₃³⋊S₃, C₃×C₃²⋊C₆, C₃×C₉⋊C₆, C₃²×C₃⋊S₃, C₃×C₃≀C₃, C₃×C₃³⋊S₃
Quotients: C₁, C₂, C₃, S₃, C₆, C₃×S₃, C₃⋊S₃, He₃⋊C₂, C₃×C₃⋊S₃, C₃³⋊S₃, C₃×He₃⋊C₂, C₃×C₃³⋊S₃

Permutation representations of C₃×C₃³⋊S₃
►On 18 points - transitive group 18T169

Generators in S₁₈

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

(7 9 8)(16 17 18)

(4 6 5)(7 8 9)(10 11 12)(16 18 17)

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

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

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

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

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

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

G:=TransitiveGroup(18,169);

►On 27 points - transitive group 27T164

Generators in S₂₇

(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)

(4 23 19)(5 24 20)(6 22 21)(10 15 18)(11 13 16)(12 14 17)

(1 8 26)(2 9 27)(3 7 25)(4 19 23)(5 20 24)(6 21 22)

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

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

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

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

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

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

G:=TransitiveGroup(27,164);

►On 27 points - transitive group 27T178

Generators in S₂₇

(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)

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

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

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

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

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

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

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

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

G:=TransitiveGroup(27,178);

39 conjugacy classes

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

39 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	3	6	6
type	+	+			+	+	+					+
image	C₁	C₂	C₃	C₆	S₃	S₃	S₃	C₃×S₃	C₃×S₃	C₃×S₃	He₃⋊C₂	C₃³⋊S₃	C₃×C₃³⋊S₃
kernel	C₃×C₃³⋊S₃	C₃×C₃≀C₃	C₃³⋊S₃	C₃≀C₃	C₃×He₃	C₃×3_-¹⁺²	C₃⁴	He₃	3_-¹⁺²	C₃³	C₃²	C₃	C₁
# reps	1	1	2	2	1	2	1	2	4	2	12	3	6

Matrix representation of C₃×C₃³⋊S₃ ►in GL₆(𝔽₁₉)

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

1	0	8	0	0	18
0	1	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	7

1	8	1	18	0	11
0	11	0	0	0	0
0	0	7	0	0	0
0	0	0	7	0	0
0	0	0	0	1	0
0	0	0	0	0	11

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

1	0	0	0	0	0
6	18	18	1	1	1
0	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0

18	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
6	18	18	1	1	1
0	1	0	0	0	0
0	0	1	0	0	0

G:=sub<GL(6,GF(19))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,8,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,18,0,0,0,0,7],[1,0,0,0,0,0,8,11,0,0,0,0,1,0,7,0,0,0,18,0,0,7,0,0,0,0,0,0,1,0,11,0,0,0,0,11],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,7,0,0,7,0,0,7,0,0,0,7,0,7,0,0,0,0,7],[1,6,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,0,1,0],[18,0,0,6,0,0,0,0,0,18,1,0,0,0,0,18,0,1,0,0,0,1,0,0,0,1,0,1,0,0,0,0,1,1,0,0] >;

C₃×C₃³⋊S₃ in GAP, Magma, Sage, TeX

C_3\times C_3^3\rtimes S_3

% in TeX

G:=Group("C3xC3^3:S3");

// GroupNames label

G:=SmallGroup(486,165);

// by ID

G=gap.SmallGroup(486,165);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,867,303,11344,1096,11669]);

// Polycyclic

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

// generators/relations

G = C3×C33⋊S3 order 486 = 2·35

Direct product of C3 and C33⋊S3

G = C₃×C₃³⋊S₃ order 486 = 2·3⁵

Direct product of C₃ and C₃³⋊S₃