C3xC3^2:C6 - GroupNames

G = C₃×C₃²⋊C₆ order 162 = 2·3⁴

Direct product of C₃ and C₃²⋊C₆

direct product, metabelian, supersoluble, monomial

Aliases: C₃×C₃²⋊C₆, He₃⋊₄C₆, C₃³⋊₂C₆, C₃³⋊₃S₃, C₃⋊S₃⋊C₃², C₃²⋊(C₃×C₆), (C₃×He₃)⋊₁C₂, C₃²⋊₁(C₃×S₃), C₃.₂(S₃×C₃²), (C₃×C₃⋊S₃)⋊C₃, SmallGroup(162,34)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃×C₃²⋊C₆

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃×He₃ — C₃×C₃²⋊C₆

Lower central

C₃² — C₃×C₃²⋊C₆

Upper central

C₁ — C₃

Generators and relations for C₃×C₃²⋊C₆
G = < a,b,c,d | a³=b³=c³=d⁶=1, ab=ba, ac=ca, ad=da, bc=cb, dbd^-1=b^-1c^-1, dcd^-1=c^-1 >

Subgroups: 200 in 58 conjugacy classes, 20 normal (11 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₃², C₃², C₃², C₃×S₃, C₃⋊S₃, C₃×C₆, He₃, He₃, C₃³, C₃³, C₃²⋊C₆, S₃×C₃², C₃×C₃⋊S₃, C₃×He₃, C₃×C₃²⋊C₆
Quotients: C₁, C₂, C₃, S₃, C₆, C₃², C₃×S₃, C₃×C₆, C₃²⋊C₆, S₃×C₃², C₃×C₃²⋊C₆

Character table of C₃×C₃²⋊C₆

class	1	2	3A	3B	3C	3D	3E	3F	3G	3H	3I	3J	3K	3L	3M	3N	3O	3P	3Q	3R	3S	3T	6A	6B	6C	6D	6E	6F	6G	6H
size	1	9	1	1	2	2	2	3	3	3	3	3	3	6	6	6	6	6	6	6	6	6	9	9	9	9	9	9	9	9
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₃	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	ζ₃	ζ₃	ζ₃²	ζ₃²	1	1	ζ₃²	ζ₃	1	ζ₃	ζ₃²	ζ₃²	ζ₃	1	1	ζ₃	ζ₃²	ζ₃²	1	ζ₃	ζ₃	ζ₃²	1	linear of order 3
ρ₄	1	-1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	ζ₃²	1	ζ₃	1	ζ₃	ζ₃²	ζ₃²	1	1	ζ₃²	ζ₃	1	ζ₃	ζ₃	ζ₃²	ζ₆⁵	ζ₆	ζ₆⁵	ζ₆	-1	ζ₆	-1	ζ₆⁵	linear of order 6
ρ₅	1	-1	1	1	1	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃²	1	ζ₃²	1	ζ₃	ζ₃²	ζ₃	1	ζ₃	ζ₃²	-1	-1	ζ₆	ζ₆	ζ₆	ζ₆⁵	ζ₆⁵	ζ₆⁵	linear of order 6
ρ₆	1	-1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	ζ₃	ζ₃	ζ₃²	ζ₃²	1	1	ζ₃²	ζ₃	1	ζ₃	ζ₃²	ζ₃²	ζ₃	1	1	ζ₆⁵	ζ₆	ζ₆	-1	ζ₆⁵	ζ₆⁵	ζ₆	-1	linear of order 6
ρ₇	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	ζ₃²	1	ζ₃	1	ζ₃	ζ₃²	ζ₃²	1	1	ζ₃²	ζ₃	1	ζ₃	ζ₃	ζ₃²	ζ₃	ζ₃²	ζ₃	ζ₃²	1	ζ₃²	1	ζ₃	linear of order 3
ρ₈	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	ζ₃	1	ζ₃²	1	ζ₃²	ζ₃	ζ₃	1	1	ζ₃	ζ₃²	1	ζ₃²	ζ₃²	ζ₃	ζ₃²	ζ₃	ζ₃²	ζ₃	1	ζ₃	1	ζ₃²	linear of order 3
ρ₉	1	-1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	ζ₃²	ζ₃²	ζ₃	ζ₃	1	1	ζ₃	ζ₃²	1	ζ₃²	ζ₃	ζ₃	ζ₃²	1	1	ζ₆	ζ₆⁵	ζ₆⁵	-1	ζ₆	ζ₆	ζ₆⁵	-1	linear of order 6
ρ₁₀	1	-1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	1	ζ₃²	1	ζ₃	ζ₃²	ζ₃	ζ₃²	ζ₃²	1	1	1	ζ₃	ζ₃	ζ₃²	ζ₃	ζ₆⁵	ζ₆	-1	ζ₆⁵	ζ₆	-1	ζ₆⁵	ζ₆	linear of order 6
ρ₁₁	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	ζ₃²	ζ₃²	ζ₃	ζ₃	1	1	ζ₃	ζ₃²	1	ζ₃²	ζ₃	ζ₃	ζ₃²	1	1	ζ₃²	ζ₃	ζ₃	1	ζ₃²	ζ₃²	ζ₃	1	linear of order 3
ρ₁₂	1	1	1	1	1	1	1	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃	1	ζ₃	1	ζ₃²	ζ₃	ζ₃²	1	ζ₃²	ζ₃	1	1	ζ₃	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃²	linear of order 3
ρ₁₃	1	1	1	1	1	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃²	1	ζ₃²	1	ζ₃	ζ₃²	ζ₃	1	ζ₃	ζ₃²	1	1	ζ₃²	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃	linear of order 3
ρ₁₄	1	1	ζ₃	ζ₃²	ζ₃	ζ₃²	1	1	ζ₃²	1	ζ₃	ζ₃²	ζ₃	ζ₃²	ζ₃²	1	1	1	ζ₃	ζ₃	ζ₃²	ζ₃	ζ₃	ζ₃²	1	ζ₃	ζ₃²	1	ζ₃	ζ₃²	linear of order 3
ρ₁₅	1	-1	1	1	1	1	1	ζ₃²	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃	1	ζ₃	1	ζ₃²	ζ₃	ζ₃²	1	ζ₃²	ζ₃	-1	-1	ζ₆⁵	ζ₆⁵	ζ₆⁵	ζ₆	ζ₆	ζ₆	linear of order 6
ρ₁₆	1	-1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	ζ₃	1	ζ₃²	1	ζ₃²	ζ₃	ζ₃	1	1	ζ₃	ζ₃²	1	ζ₃²	ζ₃²	ζ₃	ζ₆	ζ₆⁵	ζ₆	ζ₆⁵	-1	ζ₆⁵	-1	ζ₆	linear of order 6
ρ₁₇	1	1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	1	ζ₃	1	ζ₃²	ζ₃	ζ₃²	ζ₃	ζ₃	1	1	1	ζ₃²	ζ₃²	ζ₃	ζ₃²	ζ₃²	ζ₃	1	ζ₃²	ζ₃	1	ζ₃²	ζ₃	linear of order 3
ρ₁₈	1	-1	ζ₃²	ζ₃	ζ₃²	ζ₃	1	1	ζ₃	1	ζ₃²	ζ₃	ζ₃²	ζ₃	ζ₃	1	1	1	ζ₃²	ζ₃²	ζ₃	ζ₃²	ζ₆	ζ₆⁵	-1	ζ₆	ζ₆⁵	-1	ζ₆	ζ₆⁵	linear of order 6
ρ₁₉	2	0	2	2	2	2	2	2	2	2	2	2	2	-1	-1	-1	-1	-1	-1	-1	-1	-1	0	0	0	0	0	0	0	0	orthogonal lifted from S₃
ρ₂₀	2	0	-1-√-3	-1+√-3	-1-√-3	-1+√-3	2	2	-1+√-3	2	-1-√-3	-1+√-3	-1-√-3	ζ₆⁵	ζ₆⁵	-1	-1	-1	ζ₆	ζ₆	ζ₆⁵	ζ₆	0	0	0	0	0	0	0	0	complex lifted from C₃×S₃
ρ₂₁	2	0	-1-√-3	-1+√-3	-1-√-3	-1+√-3	2	-1-√-3	-1-√-3	-1+√-3	-1+√-3	2	2	ζ₆⁵	ζ₆	-1	ζ₆	ζ₆⁵	ζ₆⁵	ζ₆	-1	-1	0	0	0	0	0	0	0	0	complex lifted from C₃×S₃
ρ₂₂	2	0	2	2	2	2	2	-1+√-3	-1-√-3	-1-√-3	-1+√-3	-1+√-3	-1-√-3	-1	ζ₆	-1	ζ₆⁵	ζ₆	ζ₆⁵	-1	ζ₆⁵	ζ₆	0	0	0	0	0	0	0	0	complex lifted from C₃×S₃
ρ₂₃	2	0	2	2	2	2	2	-1-√-3	-1+√-3	-1+√-3	-1-√-3	-1-√-3	-1+√-3	-1	ζ₆⁵	-1	ζ₆	ζ₆⁵	ζ₆	-1	ζ₆	ζ₆⁵	0	0	0	0	0	0	0	0	complex lifted from C₃×S₃
ρ₂₄	2	0	-1+√-3	-1-√-3	-1+√-3	-1-√-3	2	-1+√-3	-1+√-3	-1-√-3	-1-√-3	2	2	ζ₆	ζ₆⁵	-1	ζ₆⁵	ζ₆	ζ₆	ζ₆⁵	-1	-1	0	0	0	0	0	0	0	0	complex lifted from C₃×S₃
ρ₂₅	2	0	-1+√-3	-1-√-3	-1+√-3	-1-√-3	2	-1-√-3	2	-1+√-3	2	-1+√-3	-1-√-3	ζ₆	-1	-1	ζ₆	ζ₆⁵	-1	ζ₆⁵	ζ₆⁵	ζ₆	0	0	0	0	0	0	0	0	complex lifted from C₃×S₃
ρ₂₆	2	0	-1+√-3	-1-√-3	-1+√-3	-1-√-3	2	2	-1-√-3	2	-1+√-3	-1-√-3	-1+√-3	ζ₆	ζ₆	-1	-1	-1	ζ₆⁵	ζ₆⁵	ζ₆	ζ₆⁵	0	0	0	0	0	0	0	0	complex lifted from C₃×S₃
ρ₂₇	2	0	-1-√-3	-1+√-3	-1-√-3	-1+√-3	2	-1+√-3	2	-1-√-3	2	-1-√-3	-1+√-3	ζ₆⁵	-1	-1	ζ₆⁵	ζ₆	-1	ζ₆	ζ₆	ζ₆⁵	0	0	0	0	0	0	0	0	complex lifted from C₃×S₃
ρ₂₈	6	0	6	6	-3	-3	-3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₃²⋊C₆
ρ₂₉	6	0	-3+3√-3	-3-3√-3	^3-3√-3/₂	^3+3√-3/₂	-3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful
ρ₃₀	6	0	-3-3√-3	-3+3√-3	^3+3√-3/₂	^3-3√-3/₂	-3	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful

Permutation representations of C₃×C₃²⋊C₆
►On 18 points - transitive group 18T76

Generators in S₁₈

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

(1 8 17)(3 13 10)(4 14 11)(6 7 16)

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

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

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

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

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

G:=TransitiveGroup(18,76);

►On 18 points - transitive group 18T78

Generators in S₁₈

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

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

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

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

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

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

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

G:=TransitiveGroup(18,78);

►On 18 points - transitive group 18T81

Generators in S₁₈

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

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

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

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

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

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

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

G:=TransitiveGroup(18,81);

►On 27 points - transitive group 27T48

Generators in S₂₇

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

(1 27 24)(2 22 25)(4 20 17)(5 21 18)(7 10 13)(9 15 12)

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

(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)| (1,4,9)(2,5,7)(3,6,8)(10,22,21)(11,23,16)(12,24,17)(13,25,18)(14,26,19)(15,27,20), (1,27,24)(2,22,25)(4,20,17)(5,21,18)(7,10,13)(9,15,12), (1,27,24)(2,25,22)(3,23,26)(4,20,17)(5,18,21)(6,16,19)(7,13,10)(8,11,14)(9,15,12), (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( (1,4,9)(2,5,7)(3,6,8)(10,22,21)(11,23,16)(12,24,17)(13,25,18)(14,26,19)(15,27,20), (1,27,24)(2,22,25)(4,20,17)(5,21,18)(7,10,13)(9,15,12), (1,27,24)(2,25,22)(3,23,26)(4,20,17)(5,18,21)(6,16,19)(7,13,10)(8,11,14)(9,15,12), (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([[(1,4,9),(2,5,7),(3,6,8),(10,22,21),(11,23,16),(12,24,17),(13,25,18),(14,26,19),(15,27,20)], [(1,27,24),(2,22,25),(4,20,17),(5,21,18),(7,10,13),(9,15,12)], [(1,27,24),(2,25,22),(3,23,26),(4,20,17),(5,18,21),(6,16,19),(7,13,10),(8,11,14),(9,15,12)], [(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(27,48);

►On 27 points - transitive group 27T60

Generators in S₂₇

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

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

(1 18 21)(2 16 19)(3 20 17)(4 22 13)(5 14 23)(6 24 15)(7 10 25)(8 26 11)(9 12 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)| (1,3,2)(4,15,26)(5,10,27)(6,11,22)(7,12,23)(8,13,24)(9,14,25)(16,18,20)(17,19,21), (1,13,10)(2,8,5)(3,24,27)(4,25,18)(6,12,17)(7,21,22)(9,20,15)(11,23,19)(14,16,26), (1,18,21)(2,16,19)(3,20,17)(4,22,13)(5,14,23)(6,24,15)(7,10,25)(8,26,11)(9,12,27), (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( (1,3,2)(4,15,26)(5,10,27)(6,11,22)(7,12,23)(8,13,24)(9,14,25)(16,18,20)(17,19,21), (1,13,10)(2,8,5)(3,24,27)(4,25,18)(6,12,17)(7,21,22)(9,20,15)(11,23,19)(14,16,26), (1,18,21)(2,16,19)(3,20,17)(4,22,13)(5,14,23)(6,24,15)(7,10,25)(8,26,11)(9,12,27), (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([[(1,3,2),(4,15,26),(5,10,27),(6,11,22),(7,12,23),(8,13,24),(9,14,25),(16,18,20),(17,19,21)], [(1,13,10),(2,8,5),(3,24,27),(4,25,18),(6,12,17),(7,21,22),(9,20,15),(11,23,19),(14,16,26)], [(1,18,21),(2,16,19),(3,20,17),(4,22,13),(5,14,23),(6,24,15),(7,10,25),(8,26,11),(9,12,27)], [(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(27,60);

C₃×C₃²⋊C₆ is a maximal subgroup of
He₃⋊₅D₆ C₃.C₃≀S₃ C₃²⋊C₉⋊C₆ C₃.₃C₃≀S₃ C₃⁴⋊C₆ C₉⋊He₃⋊C₂ (C₃×He₃)⋊C₆ C₉⋊S₃⋊C₃² He₃.(C₃×S₃) C₃⁴⋊₃S₃ (C₃²×C₉)⋊S₃ C₃³⋊(C₃×S₃) He₃.C₃⋊₂C₆ He₃⋊(C₃×S₃) 3₊¹⁺⁴⋊C₂ 3_-¹⁺⁴⋊C₂
C₃×C₃²⋊C₆ is a maximal quotient of
C₃⁴⋊C₆ C₃⁴⋊S₃ C₃⁴.C₆ C₃⁴.S₃ C₉⋊He₃⋊C₂ C₃≀S₃⋊₃C₃ C₃≀C₃⋊C₆ (C₃×He₃)⋊C₆ He₃.C₃⋊C₆ C₉⋊S₃⋊C₃² He₃.(C₃×C₆) He₃.(C₃×S₃) C₃≀C₃.C₆

Matrix representation of C₃×C₃²⋊C₆ ►in GL₆(𝔽₇)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

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

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

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

C₃×C₃²⋊C₆ in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes C_6

% in TeX

G:=Group("C3xC3^2:C6");

// GroupNames label

G:=SmallGroup(162,34);

// by ID

G=gap.SmallGroup(162,34);

# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,723,728,2704]);

// Polycyclic

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

// generators/relations

Export

Character table of C₃×C₃²⋊C₆ in TeX

G = C3×C32⋊C6 order 162 = 2·34

Direct product of C3 and C32⋊C6

G = C₃×C₃²⋊C₆ order 162 = 2·3⁴

Direct product of C₃ and C₃²⋊C₆