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G = S3×C32⋊C6order 324 = 22·34

Direct product of S3 and C32⋊C6

direct product, metabelian, supersoluble, monomial

Aliases: S3×C32⋊C6, He37D6, C332D6, C323S32, C33⋊(C2×C6), (S3×C32)⋊C6, C33⋊C2⋊C6, (S3×He3)⋊1C2, C322(S3×C6), (S3×C32)⋊1S3, (C3×He3)⋊1C22, He34S31C2, (C3×C3⋊S3)⋊C6, (S3×C3⋊S3)⋊C3, C3.2(C3×S32), C3⋊S32(C3×S3), C31(C2×C32⋊C6), (C3×S3).2(C3×S3), (C3×C32⋊C6)⋊1C2, SmallGroup(324,116)

Series: Derived Chief Lower central Upper central

C1C33 — S3×C32⋊C6
C1C3C32C33C3×He3S3×He3 — S3×C32⋊C6
C33 — S3×C32⋊C6
C1

Generators and relations for S3×C32⋊C6
 G = < a,b,c,d,e | a3=b2=c3=d3=e6=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1d-1, ede-1=d-1 >

Subgroups: 676 in 100 conjugacy classes, 23 normal (all characteristic)
C1, C2 [×3], C3 [×2], C3 [×7], C22, S3, S3 [×7], C6 [×8], C32 [×3], C32 [×10], D6 [×3], C2×C6, C3×S3, C3×S3 [×10], C3⋊S3, C3⋊S3 [×5], C3×C6 [×4], He3, He3 [×3], C33 [×2], C33, S32 [×2], S3×C6 [×2], C2×C3⋊S3, C32⋊C6, C32⋊C6 [×3], C2×He3, S3×C32 [×2], S3×C32 [×2], C3×C3⋊S3, C3×C3⋊S3, C33⋊C2, C3×He3, C2×C32⋊C6, C3×S32, S3×C3⋊S3, C3×C32⋊C6, S3×He3, He34S3, S3×C32⋊C6
Quotients: C1, C2 [×3], C3, C22, S3 [×2], C6 [×3], D6 [×2], C2×C6, C3×S3 [×2], S32, S3×C6 [×2], C32⋊C6, C2×C32⋊C6, C3×S32, S3×C32⋊C6

Character table of S3×C32⋊C6

 class 12A2B2C3A3B3C3D3E3F3G3H3I3J3K3L3M6A6B6C6D6E6F6G6H6I6J6K6L6M
 size 139272233466666121212699991818181818182727
ρ1111111111111111111111111111111    trivial
ρ21-1-111111111111111-1-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ311-1-111111111111111-1-111-1-11-111-1-1    linear of order 2
ρ41-11-11111111111111-111-1-111-11-1-1-1-1    linear of order 2
ρ51-11-111ζ3ζ321ζ32ζ32ζ31ζ3ζ31ζ32-1ζ3ζ32ζ65ζ61ζ3ζ65ζ32ζ6-1ζ65ζ6    linear of order 6
ρ611-1-111ζ32ζ31ζ3ζ3ζ321ζ32ζ321ζ31ζ6ζ65ζ32ζ3-1ζ6ζ32ζ65ζ31ζ6ζ65    linear of order 6
ρ711-1-111ζ3ζ321ζ32ζ32ζ31ζ3ζ31ζ321ζ65ζ6ζ3ζ32-1ζ65ζ3ζ6ζ321ζ65ζ6    linear of order 6
ρ81-11-111ζ32ζ31ζ3ζ3ζ321ζ32ζ321ζ3-1ζ32ζ3ζ6ζ651ζ32ζ6ζ3ζ65-1ζ6ζ65    linear of order 6
ρ91-1-1111ζ32ζ31ζ3ζ3ζ321ζ32ζ321ζ3-1ζ6ζ65ζ6ζ65-1ζ6ζ6ζ65ζ65-1ζ32ζ3    linear of order 6
ρ101-1-1111ζ3ζ321ζ32ζ32ζ31ζ3ζ31ζ32-1ζ65ζ6ζ65ζ6-1ζ65ζ65ζ6ζ6-1ζ3ζ32    linear of order 6
ρ11111111ζ3ζ321ζ32ζ32ζ31ζ3ζ31ζ321ζ3ζ32ζ3ζ321ζ3ζ3ζ32ζ321ζ3ζ32    linear of order 3
ρ12111111ζ32ζ31ζ3ζ3ζ321ζ32ζ321ζ31ζ32ζ3ζ32ζ31ζ32ζ32ζ3ζ31ζ32ζ3    linear of order 3
ρ132200222222-1-1-12-1-1-12002200-10-1-100    orthogonal lifted from S3
ρ1420-20-1222-1-1222-1-1-1-10-2-20011010000    orthogonal lifted from D6
ρ152020-1222-1-1222-1-1-1-102200-1-10-10000    orthogonal lifted from S3
ρ162-200222222-1-1-12-1-1-1-200-2-200101100    orthogonal lifted from D6
ρ172-20022-1--3-1+-32-1+-3ζ65ζ6-1-1--3ζ6-1ζ65-2001+-31--300ζ320ζ3100    complex lifted from S3×C6
ρ182-20022-1+-3-1--32-1--3ζ6ζ65-1-1+-3ζ65-1ζ6-2001--31+-300ζ30ζ32100    complex lifted from S3×C6
ρ192020-12-1+-3-1--3-1ζ6-1--3-1+-32ζ65ζ65-1ζ60-1+-3-1--300-1ζ650ζ60000    complex lifted from C3×S3
ρ202020-12-1--3-1+-3-1ζ65-1+-3-1--32ζ6ζ6-1ζ650-1--3-1+-300-1ζ60ζ650000    complex lifted from C3×S3
ρ2120-20-12-1+-3-1--3-1ζ6-1--3-1+-32ζ65ζ65-1ζ601--31+-3001ζ30ζ320000    complex lifted from S3×C6
ρ22220022-1+-3-1--32-1--3ζ6ζ65-1-1+-3ζ65-1ζ6200-1+-3-1--300ζ650ζ6-100    complex lifted from C3×S3
ρ2320-20-12-1--3-1+-3-1ζ65-1+-3-1--32ζ6ζ6-1ζ6501+-31--3001ζ320ζ30000    complex lifted from S3×C6
ρ24220022-1--3-1+-32-1+-3ζ65ζ6-1-1--3ζ6-1ζ65200-1--3-1+-300ζ60ζ65-100    complex lifted from C3×S3
ρ254000-2444-2-2-2-2-2-21110000000000000    orthogonal lifted from S32
ρ264000-24-2+2-3-2-2-3-21+-31+-31--3-21--3ζ31ζ320000000000000    complex lifted from C3×S32
ρ274000-24-2-2-3-2+2-3-21--31--31+-3-21+-3ζ321ζ30000000000000    complex lifted from C3×S32
ρ2866006-300-300000000-3000000000000    orthogonal lifted from C32⋊C6
ρ296-6006-300-3000000003000000000000    orthogonal lifted from C2×C32⋊C6
ρ3012000-6-6003000000000000000000000    orthogonal faithful

Permutation representations of S3×C32⋊C6
On 18 points - transitive group 18T121
Generators in S18
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)
(1 2)(3 6)(4 5)(7 14)(8 15)(9 16)(10 17)(11 18)(12 13)
(1 9 13)(2 16 12)(3 11 17)(4 14 8)(5 7 15)(6 18 10)
(1 3 5)(2 6 4)(7 9 11)(8 12 10)(13 17 15)(14 16 18)
(1 2)(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,121);

On 18 points - transitive group 18T124
Generators in S18
(1 15 12)(2 16 7)(3 17 8)(4 18 9)(5 13 10)(6 14 11)
(1 4)(2 5)(3 6)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)
(1 12 15)(2 7 16)(4 18 9)(5 13 10)
(1 12 15)(2 16 7)(3 8 17)(4 18 9)(5 10 13)(6 14 11)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,124);

On 27 points - transitive group 27T115
Generators in S27
(1 3 2)(4 9 7)(5 8 6)(10 25 19)(11 26 20)(12 27 21)(13 22 16)(14 23 17)(15 24 18)
(1 3)(4 9)(5 8)(16 22)(17 23)(18 24)(19 25)(20 26)(21 27)
(1 27 24)(2 12 15)(3 21 18)(4 25 26)(5 23 22)(6 14 13)(7 10 11)(8 17 16)(9 19 20)
(1 5 4)(2 6 7)(3 8 9)(10 12 14)(11 15 13)(16 20 18)(17 19 21)(22 26 24)(23 25 27)
(4 5)(6 7)(8 9)(10 11 12 13 14 15)(16 17 18 19 20 21)(22 23 24 25 26 27)

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

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

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

G:=TransitiveGroup(27,115);

On 27 points - transitive group 27T119
Generators in S27
(1 8 4)(2 9 5)(3 7 6)(10 23 16)(11 24 17)(12 25 18)(13 26 19)(14 27 20)(15 22 21)
(4 8)(5 9)(6 7)(10 23)(11 24)(12 25)(13 26)(14 27)(15 22)
(1 16 19)(2 17 20)(4 23 26)(5 24 27)(8 10 13)(9 11 14)
(1 16 19)(2 20 17)(3 18 21)(4 23 26)(5 27 24)(6 25 22)(7 12 15)(8 10 13)(9 14 11)
(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)

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

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

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

G:=TransitiveGroup(27,119);

Matrix representation of S3×C32⋊C6 in GL10(𝔽7)

6100000000
6000000000
0061000000
0060000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
0100000000
1000000000
0001000000
0010000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
0010000000
0001000000
6060000000
0606000000
0000100000
0000060001
0000001000
0000000610
0000000600
0000060000
,
1000000000
0100000000
0010000000
0001000000
0000601000
0000060001
0000600000
0000000060
0000000160
0000060000
,
4000000000
0400000000
3030000000
0303000000
0000000100
0000001000
0000000010
0000010000
0000000001
0000100000

G:=sub<GL(10,GF(7))| [6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,0,6,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,6,0,0,0,0,0,0,0,0,1,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,6,0,6,0,0,0,0,0,0,0,0,6,0,0,0,6,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,0,0,0,0],[4,0,3,0,0,0,0,0,0,0,0,4,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0] >;

S3×C32⋊C6 in GAP, Magma, Sage, TeX

S_3\times C_3^2\rtimes C_6
% in TeX

G:=Group("S3xC3^2:C6");
// GroupNames label

G:=SmallGroup(324,116);
// by ID

G=gap.SmallGroup(324,116);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,297,735,2164,3899]);
// Polycyclic

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

Export

Character table of S3×C32⋊C6 in TeX

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