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G = C62:S3order 216 = 23·33

4th semidirect product of C62 and S3 acting faithfully

non-abelian, soluble, monomial

Aliases: C62:4S3, C32:1S4, C3:S4:C3, (C3xA4):C6, C3.3(C3xS4), C32:A4:1C2, C22:(C32:C6), (C2xC6).4(C3xS3), SmallGroup(216,92)

Series: Derived Chief Lower central Upper central

C1C22C3xA4 — C62:S3
C1C22C2xC6C3xA4C32:A4 — C62:S3
C3xA4 — 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 >

Subgroups: 270 in 45 conjugacy classes, 10 normal (all characteristic)
Quotients: C1, C2, C3, S3, C6, C3xS3, S4, C32:C6, C3xS4, C62:S3
3C2
18C2
3C3
12C3
24C3
9C4
9C22
3C6
3C6
6C6
6S3
18C6
36S3
4C32
8C32
9D4
3D6
3C2xC6
3Dic3
3A4
6A4
9C2xC6
9C12
3C3xC6
4C3:S3
6C3xS3
4He3
3C3:D4
9C3xD4
9S4
2C3xA4
3C3xDic3
3S3xC6
4C32:C6
3C3xC3: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 C3xS3
ρ92202-1--3-1+-3ζ6ζ65-10-1--3-1+-3-1+-32-1--30000    complex lifted from C3xS3
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 C3xS4
ρ133-113-3-3-3/2-3+3-3/2000-1ζ6ζ65ζ65-1ζ6ζ3ζ32ζ65ζ6    complex lifted from C3xS4
ρ143-1-13-3+3-3/2-3-3-3/20001ζ65ζ6ζ6-1ζ65ζ6ζ65ζ32ζ3    complex lifted from C3xS4
ρ153-1-13-3-3-3/2-3+3-3/20001ζ6ζ65ζ65-1ζ6ζ65ζ6ζ3ζ32    complex lifted from C3xS4
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 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 8)(2 12 17)(3 15 7)(4 10 16)(5 13 9)(6 11 18)
(1 2)(3 6)(4 5)(7 11)(8 12)(9 10)(13 16)(14 17)(15 18)

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

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

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

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

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

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

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

G:=TransitiveGroup(18,99);

C62:S3 is a maximal subgroup of   C62:5D6
C62:S3 is a maximal quotient of   C32:CSU2(F3)  C32:2GL2(F3)  C62:5Dic3

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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