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## G = S3×C32⋊C4order 216 = 23·33

### Direct product of S3 and C32⋊C4

Aliases: S3×C32⋊C4, C33⋊(C2×C4), (S3×C32)⋊C4, C33⋊C2⋊C4, C3⋊S3.4D6, C325(C4×S3), C33⋊C41C2, (S3×C3⋊S3).C2, C31(C2×C32⋊C4), (C3×C32⋊C4)⋊2C2, (C3×C3⋊S3).3C22, SmallGroup(216,156)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — S3×C32⋊C4
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — S3×C3⋊S3 — S3×C32⋊C4
 Lower central C33 — S3×C32⋊C4
 Upper central C1

Generators and relations for S3×C32⋊C4
G = < a,b,c,d,e | a3=b2=c3=d3=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 468 in 60 conjugacy classes, 14 normal (all characteristic)
C1, C2 [×3], C3, C3 [×4], C4 [×2], C22, S3, S3 [×7], C6 [×3], C2×C4, C32, C32 [×4], Dic3, C12, D6 [×3], C3×S3 [×4], C3⋊S3, C3⋊S3 [×5], C3×C6, C4×S3, C33, C32⋊C4, C32⋊C4, S32 [×2], C2×C3⋊S3, S3×C32, C3×C3⋊S3, C33⋊C2, C2×C32⋊C4, C3×C32⋊C4, C33⋊C4, S3×C3⋊S3, S3×C32⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D6, C4×S3, C32⋊C4, C2×C32⋊C4, S3×C32⋊C4

Character table of S3×C32⋊C4

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 12A 12B size 1 3 9 27 2 4 4 8 8 9 9 27 27 12 12 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ4 1 -1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ5 1 1 -1 -1 1 1 1 1 1 i -i -i i 1 1 -1 -i i linear of order 4 ρ6 1 -1 -1 1 1 1 1 1 1 i -i i -i -1 -1 -1 -i i linear of order 4 ρ7 1 -1 -1 1 1 1 1 1 1 -i i -i i -1 -1 -1 i -i linear of order 4 ρ8 1 1 -1 -1 1 1 1 1 1 -i i i -i 1 1 -1 i -i linear of order 4 ρ9 2 0 2 0 -1 2 2 -1 -1 2 2 0 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ10 2 0 2 0 -1 2 2 -1 -1 -2 -2 0 0 0 0 -1 1 1 orthogonal lifted from D6 ρ11 2 0 -2 0 -1 2 2 -1 -1 -2i 2i 0 0 0 0 1 -i i complex lifted from C4×S3 ρ12 2 0 -2 0 -1 2 2 -1 -1 2i -2i 0 0 0 0 1 i -i complex lifted from C4×S3 ρ13 4 4 0 0 4 -2 1 1 -2 0 0 0 0 1 -2 0 0 0 orthogonal lifted from C32⋊C4 ρ14 4 -4 0 0 4 1 -2 -2 1 0 0 0 0 2 -1 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ15 4 -4 0 0 4 -2 1 1 -2 0 0 0 0 -1 2 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ16 4 4 0 0 4 1 -2 -2 1 0 0 0 0 -2 1 0 0 0 orthogonal lifted from C32⋊C4 ρ17 8 0 0 0 -4 -4 2 -1 2 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ18 8 0 0 0 -4 2 -4 2 -1 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of S3×C32⋊C4
On 12 points - transitive group 12T119
Generators in S12
(1 8 9)(2 5 10)(3 6 11)(4 7 12)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)
(1 8 9)(3 11 6)
(1 8 9)(2 10 5)(3 11 6)(4 7 12)
(1 2 3 4)(5 6 7 8)(9 10 11 12)

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

G:=Group( (1,8,9)(2,5,10)(3,6,11)(4,7,12), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11), (1,8,9)(3,11,6), (1,8,9)(2,10,5)(3,11,6)(4,7,12), (1,2,3,4)(5,6,7,8)(9,10,11,12) );

G=PermutationGroup([(1,8,9),(2,5,10),(3,6,11),(4,7,12)], [(1,3),(2,4),(5,12),(6,9),(7,10),(8,11)], [(1,8,9),(3,11,6)], [(1,8,9),(2,10,5),(3,11,6),(4,7,12)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)])

G:=TransitiveGroup(12,119);

On 18 points - transitive group 18T95
Generators in S18
(1 4 6)(2 3 5)(7 12 17)(8 13 18)(9 14 15)(10 11 16)
(1 4)(2 3)(11 16)(12 17)(13 18)(14 15)
(1 14 12)(2 11 13)(3 16 18)(4 15 17)(5 10 8)(6 9 7)
(1 12 14)(4 17 15)(6 7 9)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)

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

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

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

G:=TransitiveGroup(18,95);

On 24 points - transitive group 24T559
Generators in S24
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(1 17 16)(2 13 18)(3 14 19)(4 20 15)(5 10 23)(6 11 24)(7 21 12)(8 22 9)
(2 18 13)(4 15 20)(5 23 10)(7 12 21)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)

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

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

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

G:=TransitiveGroup(24,559);

On 27 points - transitive group 27T85
Generators in S27
(1 2 3)(4 16 25)(5 17 26)(6 18 27)(7 19 24)(8 21 13)(9 22 14)(10 23 15)(11 20 12)
(2 3)(4 25)(5 26)(6 27)(7 24)(8 21)(9 22)(10 23)(11 20)
(1 15 13)(2 10 8)(3 23 21)(4 7 22)(5 6 20)(9 25 24)(11 26 27)(12 17 18)(14 16 19)
(1 16 18)(2 25 27)(3 4 6)(5 21 22)(7 20 23)(8 9 26)(10 24 11)(12 15 19)(13 14 17)
(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)| (1,2,3)(4,16,25)(5,17,26)(6,18,27)(7,19,24)(8,21,13)(9,22,14)(10,23,15)(11,20,12), (2,3)(4,25)(5,26)(6,27)(7,24)(8,21)(9,22)(10,23)(11,20), (1,15,13)(2,10,8)(3,23,21)(4,7,22)(5,6,20)(9,25,24)(11,26,27)(12,17,18)(14,16,19), (1,16,18)(2,25,27)(3,4,6)(5,21,22)(7,20,23)(8,9,26)(10,24,11)(12,15,19)(13,14,17), (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( (1,2,3)(4,16,25)(5,17,26)(6,18,27)(7,19,24)(8,21,13)(9,22,14)(10,23,15)(11,20,12), (2,3)(4,25)(5,26)(6,27)(7,24)(8,21)(9,22)(10,23)(11,20), (1,15,13)(2,10,8)(3,23,21)(4,7,22)(5,6,20)(9,25,24)(11,26,27)(12,17,18)(14,16,19), (1,16,18)(2,25,27)(3,4,6)(5,21,22)(7,20,23)(8,9,26)(10,24,11)(12,15,19)(13,14,17), (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([(1,2,3),(4,16,25),(5,17,26),(6,18,27),(7,19,24),(8,21,13),(9,22,14),(10,23,15),(11,20,12)], [(2,3),(4,25),(5,26),(6,27),(7,24),(8,21),(9,22),(10,23),(11,20)], [(1,15,13),(2,10,8),(3,23,21),(4,7,22),(5,6,20),(9,25,24),(11,26,27),(12,17,18),(14,16,19)], [(1,16,18),(2,25,27),(3,4,6),(5,21,22),(7,20,23),(8,9,26),(10,24,11),(12,15,19),(13,14,17)], [(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(27,85);

S3×C32⋊C4 is a maximal quotient of   D6⋊(C32⋊C4)  C33⋊(C4⋊C4)  C335(C2×C8)  C33⋊M4(2)  C332M4(2)

Polynomial with Galois group S3×C32⋊C4 over ℚ
actionf(x)Disc(f)
12T119x12-x9-4x6+4x3+1318·59

Matrix representation of S3×C32⋊C4 in GL6(𝔽13)

 12 1 0 0 0 0 12 0 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 12 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 12 1 12 1 0 0 0 12 0 0 0 0 1 12 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 5 0 0 5 0 0 0 0 0 0 0 8 0

G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,1,0,0,1,1,12,12,0,0,0,12,0,0,0,0,1,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,5,0,0] >;

S3×C32⋊C4 in GAP, Magma, Sage, TeX

S_3\times C_3^2\rtimes C_4
% in TeX

G:=Group("S3xC3^2:C4");
// GroupNames label

G:=SmallGroup(216,156);
// by ID

G=gap.SmallGroup(216,156);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-3,31,489,111,490,376,5189]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^3=e^4=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,e*d*e^-1=c*d=d*c,e*c*e^-1=c^-1*d>;
// generators/relations

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