C3xC3^2:D6 - GroupNames

G = C₃×C₃²⋊D₆ order 324 = 2²·3⁴

Direct product of C₃ and C₃²⋊D₆

direct product, non-abelian, supersoluble, monomial

Aliases: C₃×C₃²⋊D₆, C₃³⋊₃D₆, C₃²⋊C₆⋊C₆, C₃²⋊(S₃×C₆), He₃⋊₂(C₂×C₆), C₃².₁₂S₃², He₃⋊C₂⋊₃C₆, (C₃×He₃)⋊₂C₂², C₃⋊S₃⋊(C₃×S₃), C₃.₄(C₃×S₃²), (C₃×C₃⋊S₃)⋊₁S₃, (C₃×C₃²⋊C₆)⋊₂C₂, (C₃×He₃⋊C₂)⋊₁C₂, SmallGroup(324,117)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — He₃ — C₃×C₃²⋊D₆

Chief series

C₁ — C₃ — C₃² — He₃ — C₃×He₃ — C₃×C₃²⋊C₆ — C₃×C₃²⋊D₆

Lower central

He₃ — C₃×C₃²⋊D₆

Upper central

C₁ — C₃

Generators and relations for C₃×C₃²⋊D₆
G = < a,b,c,d,e | a³=b³=c³=d⁶=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=ebe=b^-1c^-1, dcd^-1=c^-1, ce=ec, ede=d^-1 >

Subgroups: 546 in 103 conjugacy classes, 22 normal (10 characteristic)
C₁, C₂, C₃, C₃, C₂², S₃, C₆, C₃², C₃², C₃², D₆, C₂×C₆, C₃×S₃, C₃⋊S₃, C₃×C₆, He₃, He₃, C₃³, C₃³, S₃², S₃×C₆, C₃²⋊C₆, C₃²⋊C₆, He₃⋊C₂, S₃×C₃², C₃×C₃⋊S₃, C₃×He₃, C₃²⋊D₆, C₃×S₃², C₃×C₃²⋊C₆, C₃×He₃⋊C₂, C₃×C₃²⋊D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₆, C₂×C₆, C₃×S₃, S₃², S₃×C₆, C₃²⋊D₆, C₃×S₃², C₃×C₃²⋊D₆

Permutation representations of C₃×C₃²⋊D₆
►On 18 points - transitive group 18T118

Generators in S₁₈

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

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

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

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

(1 3)(4 6)(7 9)(10 12)(13 17)(14 16)

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

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

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

G:=TransitiveGroup(18,118);

►On 18 points - transitive group 18T126

Generators in S₁₈

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

(2 7 15)(3 8 16)(5 18 10)(6 13 11)

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

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

(1 3)(4 6)(8 12)(9 11)(13 17)(14 16)

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

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

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

G:=TransitiveGroup(18,126);

►On 27 points - transitive group 27T125

Generators in S₂₇

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

(1 16 19)(3 21 18)(4 13 10)(6 12 15)(7 22 25)(9 27 24)

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

(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 2)(4 5)(7 8)(10 14)(11 13)(16 20)(17 19)(22 26)(23 25)

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

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

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

G:=TransitiveGroup(27,125);

33 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	3F	···	3K	3L	3M	3N	6A	···	6F	6G	···	6O
order	1	2	2	2	3	3	3	3	3	3	···	3	3	3	3	6	···	6	6	···	6
size	1	9	9	9	1	1	2	2	2	6	···	6	12	12	12	9	···	9	18	···	18

33 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	4	4	6	6
type	+	+	+				+	+			+		+
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	D₆	C₃×S₃	S₃×C₆	S₃²	C₃×S₃²	C₃²⋊D₆	C₃×C₃²⋊D₆
kernel	C₃×C₃²⋊D₆	C₃×C₃²⋊C₆	C₃×He₃⋊C₂	C₃²⋊D₆	C₃²⋊C₆	He₃⋊C₂	C₃×C₃⋊S₃	C₃³	C₃⋊S₃	C₃²	C₃²	C₃	C₃	C₁
# reps	1	2	1	2	4	2	2	2	4	4	1	2	2	4

Matrix representation of C₃×C₃²⋊D₆ ►in GL₆(𝔽₇)

2	0	0	0	0	0
0	2	0	0	0	0
0	0	2	0	0	0
0	0	0	2	0	0
0	0	0	0	2	0
0	0	0	0	0	2

0	0	1	0	0	0
1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	0	1
0	0	0	1	0	0
0	0	0	0	1	0

4	0	0	0	0	0
0	4	0	0	0	0
0	0	4	0	0	0
0	0	0	2	0	0
0	0	0	0	2	0
0	0	0	0	0	2

0	0	0	0	2	0
0	0	0	4	0	0
0	0	0	0	0	1
0	0	2	0	0	0
0	1	0	0	0	0
4	0	0	0	0	0

0	0	2	0	0	0
0	1	0	0	0	0
4	0	0	0	0	0
0	0	0	0	2	0
0	0	0	4	0	0
0	0	0	0	0	1

G:=sub<GL(6,GF(7))| [2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,2,0,0,0,4,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0],[0,0,4,0,0,0,0,1,0,0,0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,2,0,0,0,0,0,0,0,1] >;

C₃×C₃²⋊D₆ in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes D_6

% in TeX

G:=Group("C3xC3^2:D6");

// GroupNames label

G:=SmallGroup(324,117);

// by ID

G=gap.SmallGroup(324,117);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,297,2164,382,7781,3899]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^6=e^2=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=e*b*e=b^-1*c^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = C3×C32⋊D6 order 324 = 22·34

Direct product of C3 and C32⋊D6

G = C₃×C₃²⋊D₆ order 324 = 2²·3⁴

Direct product of C₃ and C₃²⋊D₆