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G = C62⋊S3order 216 = 23·33

4th semidirect product of C62 and S3 acting faithfully

non-abelian, soluble, monomial

Aliases: C624S3, C321S4, C3⋊S4⋊C3, (C3×A4)⋊C6, C3.3(C3×S4), C32⋊A41C2, C22⋊(C32⋊C6), (C2×C6).4(C3×S3), SmallGroup(216,92)

Series: Derived Chief Lower central Upper central

C1C22C3×A4 — C62⋊S3
C1C22C2×C6C3×A4C32⋊A4 — C62⋊S3
C3×A4 — C62⋊S3
C1

Generators and relations for C62⋊S3
 G = < a,b,c,d,e,f | a3=b3=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, eae-1=ab-1, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

3C2
18C2
3C3
12C3
24C3
9C4
9C22
3C6
3C6
6C6
6S3
18C6
36S3
4C32
8C32
9D4
3D6
3C2×C6
3Dic3
3A4
6A4
9C2×C6
9C12
3C3×C6
4C3⋊S3
6C3×S3
4He3
3C3⋊D4
9C3×D4
9S4
2C3×A4
3C3×Dic3
3S3×C6
4C32⋊C6
3C3×C3⋊D4

Character table of C62⋊S3

 class 12A2B3A3B3C3D3E3F46A6B6C6D6E6F6G12A12B
 size 1318233242424183366618181818
ρ11111111111111111111    trivial
ρ211-1111111-111111-1-1-1-1    linear of order 2
ρ311-11ζ3ζ32ζ3ζ321-1ζ3ζ32ζ321ζ3ζ6ζ65ζ6ζ65    linear of order 6
ρ41111ζ32ζ3ζ32ζ311ζ32ζ3ζ31ζ32ζ3ζ32ζ3ζ32    linear of order 3
ρ51111ζ3ζ32ζ3ζ3211ζ3ζ32ζ321ζ3ζ32ζ3ζ32ζ3    linear of order 3
ρ611-11ζ32ζ3ζ32ζ31-1ζ32ζ3ζ31ζ32ζ65ζ6ζ65ζ6    linear of order 6
ρ7220222-1-1-10222220000    orthogonal lifted from S3
ρ82202-1+-3-1--3ζ65ζ6-10-1+-3-1--3-1--32-1+-30000    complex lifted from C3×S3
ρ92202-1--3-1+-3ζ6ζ65-10-1--3-1+-3-1+-32-1--30000    complex lifted from C3×S3
ρ103-1-13330001-1-1-1-1-1-1-111    orthogonal lifted from S4
ρ113-11333000-1-1-1-1-1-111-1-1    orthogonal lifted from S4
ρ123-113-3+3-3/2-3-3-3/2000-1ζ65ζ6ζ6-1ζ65ζ32ζ3ζ6ζ65    complex lifted from C3×S4
ρ133-113-3-3-3/2-3+3-3/2000-1ζ6ζ65ζ65-1ζ6ζ3ζ32ζ65ζ6    complex lifted from C3×S4
ρ143-1-13-3+3-3/2-3-3-3/20001ζ65ζ6ζ6-1ζ65ζ6ζ65ζ32ζ3    complex lifted from C3×S4
ρ153-1-13-3-3-3/2-3+3-3/20001ζ6ζ65ζ65-1ζ6ζ65ζ6ζ3ζ32    complex lifted from C3×S4
ρ166-20-300000044-21-20000    orthogonal faithful
ρ17660-3000000000-300000    orthogonal lifted from C32⋊C6
ρ186-20-3000000-2+2-3-2-2-31+-311--30000    complex faithful
ρ196-20-3000000-2-2-3-2+2-31--311+-30000    complex faithful

Permutation representations of C62⋊S3
On 18 points - transitive group 18T97
Generators in S18
(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 3)(2 6 4)(7 9 8)(10 12 11)(13 14 15)(16 17 18)
(1 2)(3 4)(5 6)(7 12)(8 10)(9 11)
(7 12)(8 10)(9 11)(13 16)(14 17)(15 18)
(1 7 13)(2 12 16)(3 8 15)(4 10 18)(5 9 14)(6 11 17)
(1 2)(3 6)(4 5)(7 16)(8 17)(9 18)(10 14)(11 15)(12 13)

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

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

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

G:=TransitiveGroup(18,97);

On 18 points - transitive group 18T99
Generators in S18
(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 3)(2 6 4)(7 8 9)(10 12 11)(13 15 14)(16 17 18)
(1 2)(3 4)(5 6)(10 15)(11 13)(12 14)
(7 16)(8 17)(9 18)(10 15)(11 13)(12 14)
(1 14 16)(2 12 7)(3 15 18)(4 10 9)(5 13 17)(6 11 8)
(3 5)(4 6)(7 12)(8 10)(9 11)(13 18)(14 16)(15 17)

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

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

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

G:=TransitiveGroup(18,99);

C62⋊S3 is a maximal subgroup of   C625D6
C62⋊S3 is a maximal quotient of   C32⋊CSU2(𝔽3)  C322GL2(𝔽3)  C625Dic3

Matrix representation of C62⋊S3 in GL6(ℤ)

100000
010000
000-100
001-100
0000-11
0000-10
,
-110000
-100000
00-1100
00-1000
0000-11
0000-10
,
-100000
0-10000
00-1000
000-100
000010
000001
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
000010
000001
100000
010000
001000
000100
,
0-10000
-100000
00000-1
0000-10
000-100
00-1000

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

C62⋊S3 in GAP, Magma, Sage, TeX

C_6^2\rtimes S_3
% in TeX

G:=Group("C6^2:S3");
// GroupNames label

G:=SmallGroup(216,92);
// by ID

G=gap.SmallGroup(216,92);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,218,224,867,3244,556,1949,989]);
// Polycyclic

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

Export

Subgroup lattice of C62⋊S3 in TeX
Character table of C62⋊S3 in TeX

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