C3xC3^3:C6 - GroupNames

G = C₃×C₃³⋊C₆ order 486 = 2·3⁵

Direct product of C₃ and C₃³⋊C₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃×C₃³⋊C₆, C₃⁴⋊₂C₆, C₃≀C₃⋊₈C₆, (C₃×He₃)⋊₄S₃, He₃⋊₁(C₃×S₃), C₃³⋊₇(C₃×C₆), C₃³.₅₇(C₃×S₃), C₃³⋊C₂⋊₄C₃², C₃².₁₃(S₃×C₃²), C₃².₄₂(C₃²⋊C₆), (C₃×C₃≀C₃)⋊₄C₂, C₃.₄(C₃×C₃²⋊C₆), (C₃×C₃³⋊C₂)⋊₁C₃, SmallGroup(486,116)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₃×C₃³⋊C₆

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃⁴ — C₃×C₃≀C₃ — C₃×C₃³⋊C₆

Lower central

C₃³ — C₃×C₃³⋊C₆

Upper central

C₁ — C₃

Generators and relations for C₃×C₃³⋊C₆
G = < a,b,c,d,e | a³=b³=c³=d³=e⁶=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe^-1=b^-1c^-1, cd=dc, ece^-1=c^-1d^-1, ede^-1=d^-1 >

Subgroups: 870 in 123 conjugacy classes, 22 normal (13 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₉, He₃, He₃, 3_-¹⁺², C₃³, C₃³, C₃²⋊C₆, S₃×C₃², C₃×C₃⋊S₃, C₃³⋊C₂, C₃≀C₃, C₃≀C₃, C₃×He₃, C₃×3_-¹⁺², C₃⁴, C₃³⋊C₆, C₃×C₃²⋊C₆, C₃×C₃³⋊C₂, C₃×C₃≀C₃, C₃×C₃³⋊C₆
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃×C₆, C₃²⋊C₆, S₃×C₃², C₃³⋊C₆, C₃×C₃²⋊C₆, C₃×C₃³⋊C₆

Permutation representations of C₃×C₃³⋊C₆
►On 18 points - transitive group 18T162

Generators in S₁₈

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

(1 12 15)(4 18 9)

(1 15 12)(3 8 17)(4 9 18)(6 14 11)

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

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

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

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

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

G:=TransitiveGroup(18,162);

►On 27 points - transitive group 27T199

Generators in S₂₇

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

(2 17 20)(3 21 18)(4 13 10)(6 15 12)(7 25 22)(9 27 24)

(1 19 16)(3 18 21)(4 10 13)(5 11 14)(7 22 25)(8 23 26)

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

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

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

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

G:=TransitiveGroup(27,199);

39 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	···	3Q	3R	···	3W	6A	···	6H	9A	···	9F
order	1	2	3	3	3	3	3	3	···	3	3	···	3	6	···	6	9	···	9
size	1	27	1	1	2	2	2	6	···	6	9	···	9	27	···	27	18	···	18

39 irreducible representations

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

Matrix representation of C₃×C₃³⋊C₆ ►in GL₆(𝔽₁₉)

11	0	0	0	0	0
0	11	0	0	0	0
0	0	11	0	0	0
0	0	0	11	0	0
0	0	0	0	11	0
0	0	0	0	0	11

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

7	0	0	0	0	0
0	11	0	0	0	0
0	0	1	0	0	0
12	0	0	11	0	0
0	7	0	0	7	0
0	0	0	0	0	1

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

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

G:=sub<GL(6,GF(19))| [11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[11,0,0,7,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,0,0,12,0,0,0,11,0,0,7,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1],[11,0,0,7,0,0,0,11,0,0,7,0,0,0,11,0,0,7,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,0,18,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,6,0,0,1,6,0,0,1,0,0,0,6,0,0,1,0] >;

C₃×C₃³⋊C₆ in GAP, Magma, Sage, TeX

C_3\times C_3^3\rtimes C_6

% in TeX

G:=Group("C3xC3^3:C6");

// GroupNames label

G:=SmallGroup(486,116);

// by ID

G=gap.SmallGroup(486,116);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,867,873,3244,3250,11669]);

// 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,b*d=d*b,e*b*e^-1=b^-1*c^-1,c*d=d*c,e*c*e^-1=c^-1*d^-1,e*d*e^-1=d^-1>;

// generators/relations

G = C3×C33⋊C6 order 486 = 2·35

Direct product of C3 and C33⋊C6

G = C₃×C₃³⋊C₆ order 486 = 2·3⁵

Direct product of C₃ and C₃³⋊C₆