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G = S3×C9⋊C6  order 324 = 22·34

Direct product of S3 and C9⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: S3×C9⋊C6, C33.2D6, 3- 1+2⋊3D6, C9⋊S3⋊C6, (S3×C9)⋊C6, D9⋊(C3×S3), (S3×D9)⋊C3, (C3×D9)⋊C6, C9⋊1(S3×C6), C32.5S32, (S3×C32).S3, C33.S3⋊C2, C32.10(S3×C6), (S3×3- 1+2)⋊C2, (C3×3- 1+2)⋊C22, (C3×C9⋊C6)⋊C2, (C3×C9)⋊(C2×C6), C3⋊1(C2×C9⋊C6), C3.3(C3×S32), (C3×S3).3(C3×S3), SmallGroup(324,118)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C9 — S3×C9⋊C6
Chief series
C1 — C3 — C32 — C3×C9 — C3×3- 1+2 — S3×3- 1+2 — S3×C9⋊C6
Lower central
C3×C9 — S3×C9⋊C6
Upper central
C1

Generators and relations for S3×C9⋊C6
 G = < a,b,c,d | a3=b2=c9=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c2 >

Subgroups: 424 in 80 conjugacy classes, 23 normal (all characteristic)
C1, C2, C3, C3, C22, S3, S3, C6, C9, C9, C32, C32, D6, C2×C6, D9, D9, C18, C3×S3, C3×S3, C3⋊S3, C3×C6, C3×C9, C3×C9, 3- 1+2, 3- 1+2, C33, D18, S32, S3×C6, C3×D9, S3×C9, S3×C9, C9⋊C6, C9⋊C6, C9⋊S3, C2×3- 1+2, S3×C32, S3×C32, C3×C3⋊S3, C3×3- 1+2, S3×D9, C2×C9⋊C6, C3×S32, C3×C9⋊C6, S3×3- 1+2, C33.S3, S3×C9⋊C6
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S32, S3×C6, C9⋊C6, C2×C9⋊C6, C3×S32, S3×C9⋊C6

Character table of S3×C9⋊C6

 class 12A2B2C3A3B3C3D3E3F3G6A6B6C6D6E6F6G6H6I6J9A9B9C9D9E9F18A18B18C
 size 139272233466699991818182727666121212181818
ρ1111111111111111111111111111111    trivial
ρ211-1-1111111111-11-1-1-1-1-1-1111111111    linear of order 2
ρ31-1-111111111-1-1-1-1-1-1-1-111111111-1-1-1    linear of order 2
ρ41-11-11111111-1-11-11111-1-1111111-1-1-1    linear of order 2
ρ51-11-111ζ32ζ31ζ3ζ32-1ζ65ζ3ζ6ζ321ζ3ζ32ζ65ζ61ζ3ζ32ζ31ζ32ζ6-1ζ65    linear of order 6
ρ61-1-1111ζ32ζ31ζ3ζ32-1ζ65ζ65ζ6ζ6-1ζ65ζ6ζ3ζ321ζ3ζ32ζ31ζ32ζ6-1ζ65    linear of order 6
ρ7111111ζ32ζ31ζ3ζ321ζ3ζ3ζ32ζ321ζ3ζ32ζ3ζ321ζ3ζ32ζ31ζ32ζ321ζ3    linear of order 3
ρ8111111ζ3ζ321ζ32ζ31ζ32ζ32ζ3ζ31ζ32ζ3ζ32ζ31ζ32ζ3ζ321ζ3ζ31ζ32    linear of order 3
ρ911-1-111ζ3ζ321ζ32ζ31ζ32ζ6ζ3ζ65-1ζ6ζ65ζ6ζ651ζ32ζ3ζ321ζ3ζ31ζ32    linear of order 6
ρ101-11-111ζ3ζ321ζ32ζ3-1ζ6ζ32ζ65ζ31ζ32ζ3ζ6ζ651ζ32ζ3ζ321ζ3ζ65-1ζ6    linear of order 6
ρ111-1-1111ζ3ζ321ζ32ζ3-1ζ6ζ6ζ65ζ65-1ζ6ζ65ζ32ζ31ζ32ζ3ζ321ζ3ζ65-1ζ6    linear of order 6
ρ1211-1-111ζ32ζ31ζ3ζ321ζ3ζ65ζ32ζ6-1ζ65ζ6ζ65ζ61ζ3ζ32ζ31ζ32ζ321ζ3    linear of order 6
ρ132020-1222-1-1-100202-1-1-100222-1-1-1000    orthogonal lifted from S3
ρ142-2002222222-2-20-2000000-1-1-1-1-1-1111    orthogonal lifted from D6
ρ1520-20-1222-1-1-100-20-211100222-1-1-1000    orthogonal lifted from D6
ρ16220022222222202000000-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1720-20-12-1+√-3-1-√-3-1ζ6ζ65001+√-301-√-31ζ32ζ3002-1-√-3-1+√-3ζ6-1ζ65000    complex lifted from S3×C6
ρ182020-12-1-√-3-1+√-3-1ζ65ζ600-1+√-30-1-√-3-1ζ65ζ6002-1+√-3-1-√-3ζ65-1ζ6000    complex lifted from C3×S3
ρ19220022-1+√-3-1-√-32-1-√-3-1+√-32-1-√-30-1+√-3000000-1ζ6ζ65ζ6-1ζ65ζ65-1ζ6    complex lifted from C3×S3
ρ2020-20-12-1-√-3-1+√-3-1ζ65ζ6001-√-301+√-31ζ3ζ32002-1+√-3-1-√-3ζ65-1ζ6000    complex lifted from S3×C6
ρ212-20022-1+√-3-1-√-32-1-√-3-1+√-3-21+√-301-√-3000000-1ζ6ζ65ζ6-1ζ65ζ31ζ32    complex lifted from S3×C6
ρ222-20022-1-√-3-1+√-32-1+√-3-1-√-3-21-√-301+√-3000000-1ζ65ζ6ζ65-1ζ6ζ321ζ3    complex lifted from S3×C6
ρ232020-12-1+√-3-1-√-3-1ζ6ζ6500-1-√-30-1+√-3-1ζ6ζ65002-1-√-3-1+√-3ζ6-1ζ65000    complex lifted from C3×S3
ρ24220022-1-√-3-1+√-32-1+√-3-1-√-32-1+√-30-1-√-3000000-1ζ65ζ6ζ65-1ζ6ζ6-1ζ65    complex lifted from C3×S3
ρ254000-2444-2-2-20000000000-2-2-2111000    orthogonal lifted from S32
ρ264000-24-2-2√-3-2+2√-3-21-√-31+√-30000000000-21-√-31+√-3ζ31ζ32000    complex lifted from C3×S32
ρ274000-24-2+2√-3-2-2√-3-21+√-31-√-30000000000-21+√-31-√-3ζ321ζ3000    complex lifted from C3×S32
ρ286-6006-300-3003000000000000000000    orthogonal lifted from C2×C9⋊C6
ρ2966006-300-300-3000000000000000000    orthogonal lifted from C9⋊C6
ρ3012000-6-6003000000000000000000000    orthogonal faithful

Permutation representations of S3×C9⋊C6
►On 18 points - transitive group 18T122
Generators in S18
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)

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

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

(1 16)(2 12 8 15 5 18)(3 17 6 14 9 11)(4 13)(7 10)
 
G:=sub<Sym(18)| (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18), (1,16)(2,17)(3,18)(4,10)(5,11)(6,12)(7,13)(8,14)(9,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,16)(2,12,8,15,5,18)(3,17,6,14,9,11)(4,13)(7,10)>;
 
G:=Group( (1,7,4)(2,8,5)(3,9,6)(10,13,16)(11,14,17)(12,15,18), (1,16)(2,17)(3,18)(4,10)(5,11)(6,12)(7,13)(8,14)(9,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,16)(2,12,8,15,5,18)(3,17,6,14,9,11)(4,13)(7,10) );
 
G=PermutationGroup([[(1,7,4),(2,8,5),(3,9,6),(10,13,16),(11,14,17),(12,15,18)], [(1,16),(2,17),(3,18),(4,10),(5,11),(6,12),(7,13),(8,14),(9,15)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,16),(2,12,8,15,5,18),(3,17,6,14,9,11),(4,13),(7,10)]])
 
G:=TransitiveGroup(18,122);
 

►On 27 points - transitive group 27T116
Generators in S27
(1 13 21)(2 14 22)(3 15 23)(4 16 24)(5 17 25)(6 18 26)(7 10 27)(8 11 19)(9 12 20)

(10 27)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)

(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)

(2 6 8 9 5 3)(4 7)(10 16)(11 12 17 15 14 18)(19 20 25 23 22 26)(24 27)
 
G:=sub<Sym(27)| (1,13,21)(2,14,22)(3,15,23)(4,16,24)(5,17,25)(6,18,26)(7,10,27)(8,11,19)(9,12,20), (10,27)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26), (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), (2,6,8,9,5,3)(4,7)(10,16)(11,12,17,15,14,18)(19,20,25,23,22,26)(24,27)>;
 
G:=Group( (1,13,21)(2,14,22)(3,15,23)(4,16,24)(5,17,25)(6,18,26)(7,10,27)(8,11,19)(9,12,20), (10,27)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26), (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), (2,6,8,9,5,3)(4,7)(10,16)(11,12,17,15,14,18)(19,20,25,23,22,26)(24,27) );
 
G=PermutationGroup([[(1,13,21),(2,14,22),(3,15,23),(4,16,24),(5,17,25),(6,18,26),(7,10,27),(8,11,19),(9,12,20)], [(10,27),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26)], [(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)], [(2,6,8,9,5,3),(4,7),(10,16),(11,12,17,15,14,18),(19,20,25,23,22,26),(24,27)]])
 
G:=TransitiveGroup(27,116);
 

Matrix representation of S3×C9⋊C6 ►in GL10(𝔽19)

18100000000
18000000000
00181000000
00180000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
01800000000
18000000000
00018000000
00180000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
18010000000
01801000000
18000000000
01800000000
0000000001
000018018000
0000000100
0000010000
0000100000
000001800180
,
00120000000
00012000000
12000000000
01200000000
0000100000
0000000001
000018018000
0000000010
000000018018
000001800180

G:=sub<GL(10,GF(19))| [18,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[18,0,18,0,0,0,0,0,0,0,0,18,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,1,0,0,0,0,0,0,0,0,1,0,18,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,1,0,0,0,0,0],[0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,18,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,0,18,0,0,0,0,0,1,0,0,18,0] >;
 

S3×C9⋊C6 in GAP, Magma, Sage, TeX 

S_3\times C_9\rtimes C_6
 
% in TeX
 
G:=Group("S3xC9:C6");
 
// GroupNames label
 
G:=SmallGroup(324,118);
 
// by ID
 
G=gap.SmallGroup(324,118);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,1593,735,453,2164,3899]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^2=c^9=d^6=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^2>;
 
// generators/relations
 

Export

Character table of S3×C9⋊C6 in TeX

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