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G = C62⋊C6order 216 = 23·33

3rd semidirect product of C62 and C6 acting faithfully

metabelian, soluble, monomial

Aliases: C623C6, C3⋊S3⋊A4, C32⋊(C2×A4), (C3×A4)⋊2S3, C3.3(S3×A4), C32⋊A43C2, C222(C32⋊C6), (C2×C6).7(C3×S3), (C22×C3⋊S3)⋊1C3, SmallGroup(216,99)

Series: Derived Chief Lower central Upper central

C1C62 — C62⋊C6
C1C3C32C62C32⋊A4 — C62⋊C6
C62 — C62⋊C6
C1

Generators and relations for C62⋊C6
 G = < a,b,c | a6=b6=c6=1, ab=ba, cac-1=a-1b-1, cbc-1=a3b2 >

Subgroups: 402 in 56 conjugacy classes, 11 normal (all characteristic)
C1, C2 [×3], C3, C3 [×3], C22, C22 [×2], S3 [×6], C6 [×5], C23, C32, C32 [×2], A4 [×2], D6 [×8], C2×C6, C2×C6, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C2×A4, C22×S3 [×2], He3, C3×A4, C3×A4, C2×C3⋊S3 [×2], C62, C32⋊C6, S3×A4, C22×C3⋊S3, C32⋊A4, C62⋊C6
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, C2×A4, C32⋊C6, S3×A4, C62⋊C6

Character table of C62⋊C6

 class 12A2B2C3A3B3C3D3E3F6A6B6C6D6E6F
 size 13927261212242466663636
ρ11111111111111111    trivial
ρ211-1-11111111111-1-1    linear of order 2
ρ3111111ζ32ζ3ζ3ζ321111ζ32ζ3    linear of order 3
ρ411-1-111ζ3ζ32ζ32ζ31111ζ65ζ6    linear of order 6
ρ5111111ζ3ζ32ζ32ζ31111ζ3ζ32    linear of order 3
ρ611-1-111ζ32ζ3ζ3ζ321111ζ6ζ65    linear of order 6
ρ722002-122-1-12-1-1-100    orthogonal lifted from S3
ρ822002-1-1--3-1+-3ζ65ζ62-1-1-100    complex lifted from C3×S3
ρ922002-1-1+-3-1--3ζ6ζ652-1-1-100    complex lifted from C3×S3
ρ103-1-31330000-1-1-1-100    orthogonal lifted from C2×A4
ρ113-13-1330000-1-1-1-100    orthogonal lifted from A4
ρ126-2006-30000-211100    orthogonal lifted from S3×A4
ρ136-200-3000001-2-2400    orthogonal faithful
ρ146-200-30000014-2-200    orthogonal faithful
ρ156600-300000-300000    orthogonal lifted from C32⋊C6
ρ166-200-3000001-24-200    orthogonal faithful

Permutation representations of C62⋊C6
On 18 points - transitive group 18T100
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 4 6)(2 3 5)(7 12 8 10 9 11)(13 14 15 16 17 18)
(1 18 10 4 14 12)(2 15 7 3 17 9)(5 13 8)(6 16 11)

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

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

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

G:=TransitiveGroup(18,100);

On 18 points - transitive group 18T102
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)
(1 5 4)(2 6 3)(7 11 8 12 9 10)(13 14 15 16 17 18)
(1 14 11 3 15 7)(2 17 9 4 18 12)(5 16 10 6 13 8)

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

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

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

G:=TransitiveGroup(18,102);

C62⋊C6 is a maximal subgroup of   C625D6
C62⋊C6 is a maximal quotient of   C6.(S3×A4)  Q8⋊He3⋊C2  C624C12

Matrix representation of C62⋊C6 in GL6(ℤ)

-1-10000
100000
000-100
001100
0000-10
00000-1
,
110000
-100000
00-1-100
001000
000011
0000-10
,
000010
0000-1-1
100000
-1-10000
001000
00-1-100

G:=sub<GL(6,Integers())| [-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,-1,0,0,0,0,1,0],[0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,1,-1,0,0,0,0,0,-1,0,0,0,0] >;

C62⋊C6 in GAP, Magma, Sage, TeX

C_6^2\rtimes C_6
% in TeX

G:=Group("C6^2:C6");
// GroupNames label

G:=SmallGroup(216,99);
// by ID

G=gap.SmallGroup(216,99);
# by ID

G:=PCGroup([6,-2,-3,-2,2,-3,-3,170,81,1444,1450,5189]);
// Polycyclic

G:=Group<a,b,c|a^6=b^6=c^6=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=a^3*b^2>;
// generators/relations

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Character table of C62⋊C6 in TeX

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