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## G = C2×S3≀C2order 144 = 24·32

### Direct product of C2 and S3≀C2

Aliases: C2×S3≀C2, C3⋊S3⋊D4, (C3×C6)⋊D4, S32⋊C22, C32⋊(C2×D4), C32⋊C4⋊C22, C3⋊S3.1C23, (C2×S32)⋊5C2, (C2×C32⋊C4)⋊3C2, (C2×C3⋊S3).6C22, SmallGroup(144,186)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3 — C2×S3≀C2
 Chief series C1 — C32 — C3⋊S3 — S32 — S3≀C2 — C2×S3≀C2
 Lower central C32 — C3⋊S3 — C2×S3≀C2
 Upper central C1 — C2

Generators and relations for C2×S3≀C2
G = < a,b,c,d,e | a2=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

Subgroups: 414 in 86 conjugacy classes, 21 normal (9 characteristic)
C1, C2, C2 [×6], C3 [×2], C4 [×2], C22 [×9], S3 [×8], C6 [×6], C2×C4, D4 [×4], C23 [×2], C32, D6 [×12], C2×C6 [×2], C2×D4, C3×S3 [×4], C3⋊S3 [×2], C3×C6, C22×S3 [×2], C32⋊C4 [×2], S32 [×4], S32 [×2], S3×C6 [×2], C2×C3⋊S3, S3≀C2 [×4], C2×C32⋊C4, C2×S32 [×2], C2×S3≀C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, S3≀C2, C2×S3≀C2

Character table of C2×S3≀C2

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F size 1 1 6 6 6 6 9 9 4 4 18 18 4 4 12 12 12 12 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 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ6 1 -1 -1 1 1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 linear of order 2 ρ7 1 1 1 -1 1 -1 1 1 1 1 -1 -1 1 1 -1 1 1 -1 linear of order 2 ρ8 1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ9 2 2 0 0 0 0 -2 -2 2 2 0 0 2 2 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 0 0 0 0 -2 2 2 2 0 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 0 2 0 2 0 0 -2 1 0 0 1 -2 -1 0 0 -1 orthogonal lifted from S3≀C2 ρ12 4 4 -2 0 -2 0 0 0 1 -2 0 0 -2 1 0 1 1 0 orthogonal lifted from S3≀C2 ρ13 4 -4 0 2 0 -2 0 0 -2 1 0 0 -1 2 -1 0 0 1 orthogonal faithful ρ14 4 -4 -2 0 2 0 0 0 1 -2 0 0 2 -1 0 1 -1 0 orthogonal faithful ρ15 4 -4 0 -2 0 2 0 0 -2 1 0 0 -1 2 1 0 0 -1 orthogonal faithful ρ16 4 -4 2 0 -2 0 0 0 1 -2 0 0 2 -1 0 -1 1 0 orthogonal faithful ρ17 4 4 2 0 2 0 0 0 1 -2 0 0 -2 1 0 -1 -1 0 orthogonal lifted from S3≀C2 ρ18 4 4 0 -2 0 -2 0 0 -2 1 0 0 1 -2 1 0 0 1 orthogonal lifted from S3≀C2

Permutation representations of C2×S3≀C2
On 12 points - transitive group 12T77
Generators in S12
(1 4)(2 3)(5 9)(6 10)(7 11)(8 12)
(1 6 8)(2 7 5)(3 11 9)(4 10 12)
(1 6 8)(2 5 7)(3 9 11)(4 10 12)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 3)(2 4)(5 10)(6 9)(7 12)(8 11)

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

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

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

G:=TransitiveGroup(12,77);

On 12 points - transitive group 12T78
Generators in S12
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)
(1 9 8)(3 6 11)
(2 5 10)(4 12 7)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 12)(6 11)(7 10)(8 9)

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

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

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

G:=TransitiveGroup(12,78);

On 18 points - transitive group 18T63
Generators in S18
(1 2)(3 16)(4 17)(5 18)(6 15)(7 14)(8 11)(9 12)(10 13)
(1 4 6)(2 17 15)(3 14 13)(5 11 12)(7 10 16)(8 9 18)
(1 3 5)(2 16 18)(4 14 11)(6 13 12)(7 8 17)(9 15 10)
(3 4 5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(4 6)(7 10)(8 9)(11 12)(13 14)(15 17)

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

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

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

G:=TransitiveGroup(18,63);

On 24 points - transitive group 24T261
Generators in S24
(1 11)(2 12)(3 9)(4 10)(5 24)(6 21)(7 22)(8 23)(13 17)(14 18)(15 19)(16 20)
(1 19 6)(2 20 7)(3 8 17)(4 5 18)(9 23 13)(10 24 14)(11 15 21)(12 16 22)
(1 19 6)(2 7 20)(3 8 17)(4 18 5)(9 23 13)(10 14 24)(11 15 21)(12 22 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)

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

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

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

G:=TransitiveGroup(24,261);

On 24 points - transitive group 24T262
Generators in S24
(1 18)(2 19)(3 20)(4 17)(5 13)(6 14)(7 15)(8 16)(9 22)(10 23)(11 24)(12 21)
(1 7 12)(2 9 8)(3 10 5)(4 6 11)(13 20 23)(14 24 17)(15 21 18)(16 19 22)
(1 12 7)(2 9 8)(3 5 10)(4 6 11)(13 23 20)(14 24 17)(15 18 21)(16 19 22)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4)(2 3)(5 9)(6 12)(7 11)(8 10)(13 22)(14 21)(15 24)(16 23)(17 18)(19 20)

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

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

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

G:=TransitiveGroup(24,262);

On 24 points - transitive group 24T263
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 16)(10 13)(11 14)(12 15)(17 23)(18 24)(19 21)(20 22)
(1 10 12)(2 11 9)(3 18 20)(4 17 19)(5 13 15)(6 14 16)(7 24 22)(8 23 21)
(1 10 12)(2 9 11)(3 20 18)(4 17 19)(5 13 15)(6 16 14)(7 22 24)(8 23 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)
(1 3)(2 4)(5 7)(6 8)(9 17)(10 20)(11 19)(12 18)(13 22)(14 21)(15 24)(16 23)

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

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

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

G:=TransitiveGroup(24,263);

On 24 points - transitive group 24T264
Generators in S24
(1 3)(2 4)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)(17 24)(18 21)(19 22)(20 23)
(2 22 17)(4 19 24)(5 14 10)(7 12 16)
(1 21 20)(3 18 23)(6 11 15)(8 13 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,264);

C2×S3≀C2 is a maximal subgroup of
C2.AΓL1(𝔽9)  S32⋊D4  C4⋊S3≀C2  D6≀C2  C62⋊D4
C2×S3≀C2 is a maximal quotient of
S32⋊Q8  C4.4S3≀C2  C32⋊C4⋊Q8  C32⋊D85C2  C32⋊D8⋊C2  C3⋊S3⋊D8  C32⋊Q16⋊C2  C3⋊S32SD16  C3⋊S3⋊Q16  S32⋊D4  C4⋊S3≀C2  C62.9D4  C62.12D4  C62.13D4  C62.15D4  D6≀C2  C62⋊D4

Polynomial with Galois group C2×S3≀C2 over ℚ
actionf(x)Disc(f)
12T77x12-2x10-x8-2x6+5x4-6x2+4226·56·114·5092
12T78x12-12x10+54x8-104x6+57x4+36x2-16-256·312·56

Matrix representation of C2×S3≀C2 in GL4(ℤ) generated by

 -1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1
,
 0 1 0 0 -1 -1 0 0 0 0 -1 -1 0 0 1 0
,
 -1 -1 0 0 1 0 0 0 0 0 -1 -1 0 0 1 0
,
 0 0 1 0 0 0 -1 -1 -1 0 0 0 0 -1 0 0
,
 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0
G:=sub<GL(4,Integers())| [-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,-1,0,0,1,-1,0,0,0,0,-1,1,0,0,-1,0],[-1,1,0,0,-1,0,0,0,0,0,-1,1,0,0,-1,0],[0,0,-1,0,0,0,0,-1,1,-1,0,0,0,-1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

C2×S3≀C2 in GAP, Magma, Sage, TeX

C_2\times S_3\wr C_2
% in TeX

G:=Group("C2xS3wrC2");
// GroupNames label

G:=SmallGroup(144,186);
// by ID

G=gap.SmallGroup(144,186);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,121,964,730,142,299,455]);
// Polycyclic

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

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