C3^3:1C18 - GroupNames

G = C₃³:₁C₁₈ order 486 = 2·3⁵

1^st semidirect product of C₃³ and C₁₈ acting via C₁₈/C₃=C₆

Aliases: C₃³:₁C₁₈, C₃⁴.₁C₆, C₃²:C₉:₃S₃, C₃³:C₂:₁C₉, C₃³:C₉:₁C₂, C₃².₆(S₃xC₉), C₃³.₄₉(C₃xS₃), C₃.₂(C₃²:C₁₈), C₃.₂(C₃³:C₆), C₃².₃₄(C₃²:C₆), (C₃xC₃³:C₂).₁C₃, SmallGroup(486,18)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₃³:₁C₁₈

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃⁴ — C₃³:C₉ — C₃³:₁C₁₈

Lower central

C₃³ — C₃³:₁C₁₈

Upper central

C₁ — C₃

Generators and relations for C₃³:₁C₁₈
G = < a,b,c,d | a³=b³=c³=d¹⁸=1, ab=ba, ac=ca, dad^-1=a^-1b^-1, bc=cb, dbd^-1=b^-1c^-1, dcd^-1=c^-1 >

Subgroups: 660 in 87 conjugacy classes, 13 normal (all characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², C₁₈, C₃xS₃, C₃:S₃, C₃xC₉, C₃³, C₃³, S₃xC₉, C₃xC₃:S₃, C₃³:C₂, C₃²:C₉, C₃²:C₉, C₃⁴, C₃²:C₁₈, C₃xC₃³:C₂, C₃³:C₉, C₃³:₁C₁₈
Quotients: C₁, C₂, C₃, S₃, C₆, C₉, C₁₈, C₃xS₃, S₃xC₉, C₃²:C₆, C₃²:C₁₈, C₃³:C₆, C₃³:₁C₁₈

Permutation representations of C₃³:₁C₁₈
►On 18 points - transitive group 18T159

Generators in S₁₈

(1 7 13)(2 14 8)(4 16 10)(5 11 17)

(2 8 14)(3 9 15)(5 17 11)(6 18 12)

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

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

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

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

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

G:=TransitiveGroup(18,159);

►On 27 points - transitive group 27T200

Generators in S₂₇

(1 27 18)(4 21 12)(7 15 24)

(1 18 27)(3 11 20)(4 12 21)(6 23 14)(7 24 15)(9 17 26)

(1 27 18)(2 19 10)(3 11 20)(4 21 12)(5 13 22)(6 23 14)(7 15 24)(8 25 16)(9 17 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)| (1,27,18)(4,21,12)(7,15,24), (1,18,27)(3,11,20)(4,12,21)(6,23,14)(7,24,15)(9,17,26), (1,27,18)(2,19,10)(3,11,20)(4,21,12)(5,13,22)(6,23,14)(7,15,24)(8,25,16)(9,17,26), (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( (1,27,18)(4,21,12)(7,15,24), (1,18,27)(3,11,20)(4,12,21)(6,23,14)(7,24,15)(9,17,26), (1,27,18)(2,19,10)(3,11,20)(4,21,12)(5,13,22)(6,23,14)(7,15,24)(8,25,16)(9,17,26), (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([[(1,27,18),(4,21,12),(7,15,24)], [(1,18,27),(3,11,20),(4,12,21),(6,23,14),(7,24,15),(9,17,26)], [(1,27,18),(2,19,10),(3,11,20),(4,21,12),(5,13,22),(6,23,14),(7,15,24),(8,25,16),(9,17,26)], [(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(27,200);

39 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	3F	···	3Q	6A	6B	9A	···	9F	9G	···	9L	18A	···	18F
order	1	2	3	3	3	3	3	3	···	3	6	6	9	···	9	9	···	9	18	···	18
size	1	27	1	1	2	2	2	6	···	6	27	27	9	···	9	18	···	18	27	···	27

39 irreducible representations

dim	1	1	1	1	1	1	2	2	2	6	6	6	6
type	+	+					+			+		+
image	C₁	C₂	C₃	C₆	C₉	C₁₈	S₃	C₃xS₃	S₃xC₉	C₃²:C₆	C₃²:C₁₈	C₃³:C₆	C₃³:₁C₁₈
kernel	C₃³:₁C₁₈	C₃³:C₉	C₃xC₃³:C₂	C₃⁴	C₃³:C₂	C₃³	C₃²:C₉	C₃³	C₃²	C₃²	C₃	C₃	C₁
# reps	1	1	2	2	6	6	1	2	6	1	2	3	6

Matrix representation of C₃³:₁C₁₈ ►in GL₆(F₁₉)

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

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

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

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

C₃³:₁C₁₈ in GAP, Magma, Sage, TeX

C_3^3\rtimes_1C_{18}

% in TeX

G:=Group("C3^3:1C18");

// GroupNames label

G:=SmallGroup(486,18);

// by ID

G=gap.SmallGroup(486,18);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,867,873,3244,3250,11669]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^18=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^-1,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=c^-1>;

// generators/relations

G = C33:1C18 order 486 = 2·35

1st semidirect product of C33 and C18 acting via C18/C3=C6

G = C₃³:₁C₁₈ order 486 = 2·3⁵

1^st semidirect product of C₃³ and C₁₈ acting via C₁₈/C₃=C₆