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G = C2×S3≀C2order 144 = 24·32

Direct product of C2 and S3≀C2

direct product, non-abelian, soluble, monomial, rational

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

C1C32C3⋊S3 — C2×S3≀C2
C1C32C3⋊S3S32S3≀C2 — C2×S3≀C2
C32C3⋊S3 — C2×S3≀C2
C1C2

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 12A2B2C2D2E2F2G3A3B4A4B6A6B6C6D6E6F
 size 116666994418184412121212
ρ1111111111111111111    trivial
ρ21-11-1-111-1111-1-1-1-11-11    linear of order 2
ρ311-11-111111-1-1111-1-11    linear of order 2
ρ41-1-1-1111-111-11-1-1-1-111    linear of order 2
ρ511-1-1-1-111111111-1-1-1-1    linear of order 2
ρ61-1-111-11-1111-1-1-11-11-1    linear of order 2
ρ7111-11-11111-1-111-111-1    linear of order 2
ρ81-111-1-11-111-11-1-111-1-1    linear of order 2
ρ9220000-2-22200220000    orthogonal lifted from D4
ρ102-20000-222200-2-20000    orthogonal lifted from D4
ρ1144020200-21001-2-100-1    orthogonal lifted from S3≀C2
ρ1244-20-20001-200-210110    orthogonal lifted from S3≀C2
ρ134-4020-200-2100-12-1001    orthogonal faithful
ρ144-4-2020001-2002-101-10    orthogonal faithful
ρ154-40-20200-2100-12100-1    orthogonal faithful
ρ164-420-20001-2002-10-110    orthogonal faithful
ρ17442020001-200-210-1-10    orthogonal lifted from S3≀C2
ρ18440-20-200-21001-21001    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

-1000
0-100
00-10
000-1
,
0100
-1-100
00-1-1
0010
,
-1-100
1000
00-1-1
0010
,
0010
00-1-1
-1000
0-100
,
0010
0001
1000
0100
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

Export

Character table of C2×S3≀C2 in TeX

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