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G = C3×S32  order 108 = 22·33

Direct product of C3, S3 and S3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C3×S32, C32⋊5D6, C33⋊1C22, (C3×S3)⋊C6, C3⋊S3⋊2C6, C3⋊1(S3×C6), C32⋊2(C2×C6), (S3×C32)⋊1C2, (C3×C3⋊S3)⋊1C2, SmallGroup(108,38)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×S32
Chief series
C1 — C3 — C32 — C33 — S3×C32 — C3×S32
Lower central
C32 — C3×S32
Upper central
C1 — C3

Generators and relations for C3×S32
 G = < a,b,c,d,e | a3=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 152 in 54 conjugacy classes, 20 normal (8 characteristic)
C1, C2, C3, C3, C3, C22, S3, S3, C6, C32, C32, C32, D6, C2×C6, C3×S3, C3×S3, C3⋊S3, C3×C6, C33, S32, S3×C6, S3×C32, C3×C3⋊S3, C3×S32
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S32, S3×C6, C3×S32

Character table of C3×S32

 class 12A2B2C3A3B3C3D3E3F3G3H3I3J3K6A6B6C6D6E6F6G6H6I6J6K6L
 size 133911222222444333366666699
ρ1111111111111111111111111111    trivial
ρ21-11-111111111111-11-1111-11-1-1-1-1    linear of order 2
ρ31-1-1111111111111-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ411-1-1111111111111-11-1-1-11-111-1-1    linear of order 2
ρ51-1-11ζ3ζ32ζ321ζ32ζ3ζ31ζ31ζ32ζ6ζ65ζ65ζ6-1ζ6ζ65ζ65ζ6-1ζ3ζ32    linear of order 6
ρ61-1-11ζ32ζ3ζ31ζ3ζ32ζ321ζ321ζ3ζ65ζ6ζ6ζ65-1ζ65ζ6ζ6ζ65-1ζ32ζ3    linear of order 6
ρ71-11-1ζ3ζ32ζ321ζ32ζ3ζ31ζ31ζ32ζ6ζ3ζ65ζ321ζ32ζ65ζ3ζ6-1ζ65ζ6    linear of order 6
ρ811-1-1ζ32ζ3ζ31ζ3ζ32ζ321ζ321ζ3ζ3ζ6ζ32ζ65-1ζ65ζ32ζ6ζ31ζ6ζ65    linear of order 6
ρ911-1-1ζ3ζ32ζ321ζ32ζ3ζ31ζ31ζ32ζ32ζ65ζ3ζ6-1ζ6ζ3ζ65ζ321ζ65ζ6    linear of order 6
ρ101111ζ32ζ3ζ31ζ3ζ32ζ321ζ321ζ3ζ3ζ32ζ32ζ31ζ3ζ32ζ32ζ31ζ32ζ3    linear of order 3
ρ111-11-1ζ32ζ3ζ31ζ3ζ32ζ321ζ321ζ3ζ65ζ32ζ6ζ31ζ3ζ6ζ32ζ65-1ζ6ζ65    linear of order 6
ρ121111ζ3ζ32ζ321ζ32ζ3ζ31ζ31ζ32ζ32ζ3ζ3ζ321ζ32ζ3ζ3ζ321ζ3ζ32    linear of order 3
ρ132-2002222-12-1-1-1-1-1-20-2000101100    orthogonal lifted from D6
ρ1422002222-12-1-1-1-1-1202000-10-1-100    orthogonal lifted from S3
ρ1520-2022-1-12-122-1-1-10-20-211010000    orthogonal lifted from D6
ρ16202022-1-12-122-1-1-10202-1-10-10000    orthogonal lifted from S3
ρ1720-20-1-√-3-1+√-3ζ65-1-1+√-3ζ6-1-√-32ζ6-1ζ6501+√-301-√-31ζ30ζ320000    complex lifted from S3×C6
ρ1820-20-1+√-3-1-√-3ζ6-1-1-√-3ζ65-1+√-32ζ65-1ζ601-√-301+√-31ζ320ζ30000    complex lifted from S3×C6
ρ192200-1-√-3-1+√-3-1+√-32ζ65-1-√-3ζ6-1ζ6-1ζ65-1+√-30-1-√-3000ζ60ζ65-100    complex lifted from C3×S3
ρ202020-1-√-3-1+√-3ζ65-1-1+√-3ζ6-1-√-32ζ6-1ζ650-1-√-30-1+√-3-1ζ650ζ60000    complex lifted from C3×S3
ρ212020-1+√-3-1-√-3ζ6-1-1-√-3ζ65-1+√-32ζ65-1ζ60-1+√-30-1-√-3-1ζ60ζ650000    complex lifted from C3×S3
ρ222200-1+√-3-1-√-3-1-√-32ζ6-1+√-3ζ65-1ζ65-1ζ6-1-√-30-1+√-3000ζ650ζ6-100    complex lifted from C3×S3
ρ232-200-1+√-3-1-√-3-1-√-32ζ6-1+√-3ζ65-1ζ65-1ζ61+√-301-√-3000ζ30ζ32100    complex lifted from S3×C6
ρ242-200-1-√-3-1+√-3-1+√-32ζ65-1-√-3ζ6-1ζ6-1ζ651-√-301+√-3000ζ320ζ3100    complex lifted from S3×C6
ρ25400044-2-2-2-2-2-2111000000000000    orthogonal lifted from S32
ρ264000-2+2√-3-2-2√-31+√-3-21+√-31-√-31-√-3-2ζ31ζ32000000000000    complex faithful
ρ274000-2-2√-3-2+2√-31-√-3-21-√-31+√-31+√-3-2ζ321ζ3000000000000    complex faithful

Permutation representations of C3×S32
►On 12 points - transitive group 12T70
Generators in S12
(1 2 3)(4 5 6)(7 8 9)(10 11 12)

(1 3 2)(4 5 6)(7 9 8)(10 11 12)

(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)

(1 2 3)(4 5 6)(7 9 8)(10 12 11)

(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)
 
G:=sub<Sym(12)| (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,3,2)(4,5,6)(7,9,8)(10,11,12), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (1,2,3)(4,5,6)(7,9,8)(10,12,11), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)>;
 
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12), (1,3,2)(4,5,6)(7,9,8)(10,11,12), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (1,2,3)(4,5,6)(7,9,8)(10,12,11), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) );
 
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12)], [(1,3,2),(4,5,6),(7,9,8),(10,11,12)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)], [(1,2,3),(4,5,6),(7,9,8),(10,12,11)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)]])
 
G:=TransitiveGroup(12,70);
 

►On 18 points - transitive group 18T43
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)

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

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

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

(1 16)(2 17)(3 18)(4 13)(5 14)(6 15)(7 11)(8 12)(9 10)
 
G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,14,10)(2,15,11)(3,13,12)(4,8,18)(5,9,16)(6,7,17), (1,16)(2,17)(3,18)(4,12)(5,10)(6,11)(7,15)(8,13)(9,14), (1,10,14)(2,11,15)(3,12,13)(4,8,18)(5,9,16)(6,7,17), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,11)(8,12)(9,10)>;
 
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,14,10)(2,15,11)(3,13,12)(4,8,18)(5,9,16)(6,7,17), (1,16)(2,17)(3,18)(4,12)(5,10)(6,11)(7,15)(8,13)(9,14), (1,10,14)(2,11,15)(3,12,13)(4,8,18)(5,9,16)(6,7,17), (1,16)(2,17)(3,18)(4,13)(5,14)(6,15)(7,11)(8,12)(9,10) );
 
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,14,10),(2,15,11),(3,13,12),(4,8,18),(5,9,16),(6,7,17)], [(1,16),(2,17),(3,18),(4,12),(5,10),(6,11),(7,15),(8,13),(9,14)], [(1,10,14),(2,11,15),(3,12,13),(4,8,18),(5,9,16),(6,7,17)], [(1,16),(2,17),(3,18),(4,13),(5,14),(6,15),(7,11),(8,12),(9,10)]])
 
G:=TransitiveGroup(18,43);
 

►On 18 points - transitive group 18T46
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)

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

(4 7)(5 8)(6 9)(13 16)(14 17)(15 18)

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

(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)
 
G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (4,7)(5,8)(6,9)(13,16)(14,17)(15,18), (1,2,3)(4,5,6)(7,8,9)(10,12,11)(13,15,14)(16,18,17), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)>;
 
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18), (4,7)(5,8)(6,9)(13,16)(14,17)(15,18), (1,2,3)(4,5,6)(7,8,9)(10,12,11)(13,15,14)(16,18,17), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18) );
 
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18)], [(4,7),(5,8),(6,9),(13,16),(14,17),(15,18)], [(1,2,3),(4,5,6),(7,8,9),(10,12,11),(13,15,14),(16,18,17)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18)]])
 
G:=TransitiveGroup(18,46);
 

►On 27 points - transitive group 27T36
Generators in S27
(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)

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

(4 15)(5 13)(6 14)(10 19)(11 20)(12 21)(16 24)(17 22)(18 23)

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

(7 25)(8 26)(9 27)(10 18)(11 16)(12 17)(19 23)(20 24)(21 22)
 
G:=sub<Sym(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), (1,14,6)(2,15,4)(3,13,5)(7,16,24)(8,17,22)(9,18,23)(10,19,27)(11,20,25)(12,21,26), (4,15)(5,13)(6,14)(10,19)(11,20)(12,21)(16,24)(17,22)(18,23), (1,8,26)(2,9,27)(3,7,25)(4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (7,25)(8,26)(9,27)(10,18)(11,16)(12,17)(19,23)(20,24)(21,22)>;
 
G:=Group( (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), (1,14,6)(2,15,4)(3,13,5)(7,16,24)(8,17,22)(9,18,23)(10,19,27)(11,20,25)(12,21,26), (4,15)(5,13)(6,14)(10,19)(11,20)(12,21)(16,24)(17,22)(18,23), (1,8,26)(2,9,27)(3,7,25)(4,23,19)(5,24,20)(6,22,21)(10,15,18)(11,13,16)(12,14,17), (7,25)(8,26)(9,27)(10,18)(11,16)(12,17)(19,23)(20,24)(21,22) );
 
G=PermutationGroup([[(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)], [(1,14,6),(2,15,4),(3,13,5),(7,16,24),(8,17,22),(9,18,23),(10,19,27),(11,20,25),(12,21,26)], [(4,15),(5,13),(6,14),(10,19),(11,20),(12,21),(16,24),(17,22),(18,23)], [(1,8,26),(2,9,27),(3,7,25),(4,23,19),(5,24,20),(6,22,21),(10,15,18),(11,13,16),(12,14,17)], [(7,25),(8,26),(9,27),(10,18),(11,16),(12,17),(19,23),(20,24),(21,22)]])
 
G:=TransitiveGroup(27,36);
 

C3×S32 is a maximal subgroup of   C33⋊D4

Polynomial with Galois group C3×S32 over ℚ
actionf(x)Disc(f)
12T70x12+9x6-18x3+9212·334·56

Matrix representation of C3×S32 ►in GL4(𝔽7) generated by

2000
0200
0010
0001
,
1000
0100
0001
0066
,
1000
0100
0010
0066
,
0100
6600
0010
0001
,
1000
6600
0010
0001
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,6,0,0,1,6],[1,0,0,0,0,1,0,0,0,0,1,6,0,0,0,6],[0,6,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[1,6,0,0,0,6,0,0,0,0,1,0,0,0,0,1] >;
 

C3×S32 in GAP, Magma, Sage, TeX 

C_3\times S_3^2
 
% in TeX
 
G:=Group("C3xS3^2");
 
// GroupNames label
 
G:=SmallGroup(108,38);
 
// by ID
 
G=gap.SmallGroup(108,38);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-3,-3,248,1804]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of C3×S32 in TeX

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