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G = S3×C32⋊C4order 216 = 23·33

Direct product of S3 and C32⋊C4

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

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

C1C33 — S3×C32⋊C4
C1C3C33C3×C3⋊S3S3×C3⋊S3 — S3×C32⋊C4
C33 — S3×C32⋊C4
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 12A2B2C3A3B3C3D3E4A4B4C4D6A6B6C12A12B
 size 13927244889927271212181818
ρ1111111111111111111    trivial
ρ21-11-111111-1-111-1-11-1-1    linear of order 2
ρ3111111111-1-1-1-1111-1-1    linear of order 2
ρ41-11-11111111-1-1-1-1111    linear of order 2
ρ511-1-111111i-i-ii11-1-ii    linear of order 4
ρ61-1-1111111i-ii-i-1-1-1-ii    linear of order 4
ρ71-1-1111111-ii-ii-1-1-1i-i    linear of order 4
ρ811-1-111111-iii-i11-1i-i    linear of order 4
ρ92020-122-1-1220000-1-1-1    orthogonal lifted from S3
ρ102020-122-1-1-2-20000-111    orthogonal lifted from D6
ρ1120-20-122-1-1-2i2i00001-ii    complex lifted from C4×S3
ρ1220-20-122-1-12i-2i00001i-i    complex lifted from C4×S3
ρ1344004-211-200001-2000    orthogonal lifted from C32⋊C4
ρ144-40041-2-2100002-1000    orthogonal lifted from C2×C32⋊C4
ρ154-4004-211-20000-12000    orthogonal lifted from C2×C32⋊C4
ρ16440041-2-210000-21000    orthogonal lifted from C32⋊C4
ρ178000-4-42-12000000000    orthogonal faithful
ρ188000-42-42-1000000000    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)

1210000
1200000
001000
000100
000010
000001
,
0120000
1200000
0012000
0001200
0000120
0000012
,
100000
010000
000101
00121121
0001200
0011200
,
100000
010000
0012100
0012000
0000121
0000120
,
500000
050000
000800
000005
005000
000080

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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Character table of S3×C32⋊C4 in TeX

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