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## G = C2×S32order 72 = 23·32

### Direct product of C2, S3 and S3

Aliases: C2×S32, C61D6, C32⋊C23, C3⋊S3⋊C22, (C3×C6)⋊C22, (S3×C6)⋊5C2, (C3×S3)⋊C22, C31(C22×S3), (C2×C3⋊S3)⋊4C2, SmallGroup(72,46)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×S32
G = < a,b,c,d,e | a2=b3=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 206 in 69 conjugacy classes, 28 normal (6 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C22 [×7], S3 [×4], S3 [×6], C6 [×2], C6 [×5], C23, C32, D6 [×2], D6 [×11], C2×C6 [×2], C3×S3 [×4], C3⋊S3 [×2], C3×C6, C22×S3 [×2], S32 [×4], S3×C6 [×2], C2×C3⋊S3, C2×S32
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C22×S3 [×2], S32, C2×S32

Character table of C2×S32

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 6A 6B 6C 6D 6E 6F 6G size 1 1 3 3 3 3 9 9 2 2 4 2 2 4 6 6 6 6 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 2 0 2 0 0 2 -1 -1 -1 2 -1 -1 0 -1 0 orthogonal lifted from S3 ρ10 2 -2 0 -2 0 2 0 0 2 -1 -1 1 -2 1 1 0 -1 0 orthogonal lifted from D6 ρ11 2 2 0 -2 0 -2 0 0 2 -1 -1 -1 2 -1 1 0 1 0 orthogonal lifted from D6 ρ12 2 -2 0 2 0 -2 0 0 2 -1 -1 1 -2 1 -1 0 1 0 orthogonal lifted from D6 ρ13 2 -2 -2 0 2 0 0 0 -1 2 -1 -2 1 1 0 1 0 -1 orthogonal lifted from D6 ρ14 2 2 -2 0 -2 0 0 0 -1 2 -1 2 -1 -1 0 1 0 1 orthogonal lifted from D6 ρ15 2 2 2 0 2 0 0 0 -1 2 -1 2 -1 -1 0 -1 0 -1 orthogonal lifted from S3 ρ16 2 -2 2 0 -2 0 0 0 -1 2 -1 -2 1 1 0 -1 0 1 orthogonal lifted from D6 ρ17 4 -4 0 0 0 0 0 0 -2 -2 1 2 2 -1 0 0 0 0 orthogonal faithful ρ18 4 4 0 0 0 0 0 0 -2 -2 1 -2 -2 1 0 0 0 0 orthogonal lifted from S32

Permutation representations of C2×S32
On 12 points - transitive group 12T37
Generators in S12
(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 10)(2 12)(3 11)(4 8)(5 7)(6 9)
(1 3 2)(4 6 5)(7 8 9)(10 11 12)
(1 7)(2 8)(3 9)(4 12)(5 10)(6 11)

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

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

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

G:=TransitiveGroup(12,37);

On 18 points - transitive group 18T29
Generators in S18
(1 9)(2 7)(3 8)(4 12)(5 10)(6 11)(13 17)(14 18)(15 16)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 9)(2 8)(3 7)(4 11)(5 10)(6 12)(13 17)(14 16)(15 18)
(1 13 10)(2 14 11)(3 15 12)(4 8 16)(5 9 17)(6 7 18)
(4 16)(5 17)(6 18)(10 13)(11 14)(12 15)

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

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

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

G:=TransitiveGroup(18,29);

On 24 points - transitive group 24T73
Generators in S24
(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(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 10)(2 12)(3 11)(4 8)(5 7)(6 9)(13 23)(14 22)(15 24)(16 20)(17 19)(18 21)
(1 3 2)(4 6 5)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 21 20)(22 24 23)
(1 17)(2 18)(3 16)(4 13)(5 14)(6 15)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)

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

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

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

G:=TransitiveGroup(24,73);

C2×S32 is a maximal subgroup of
S32⋊C4  D6⋊D6  Dic3⋊D6
C2×S32 is a maximal quotient of
D125S3  D12⋊S3  Dic3.D6  D6.D6  D6.6D6  D6⋊D6  D6.3D6  D6.4D6  Dic3⋊D6

Polynomial with Galois group C2×S32 over ℚ
actionf(x)Disc(f)
12T37x12-32x10+384x8-2127x6+5360x4-5056x2+256252·36·74·376·476

Matrix representation of C2×S32 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
,
 0 0 -1 0 0 0 0 -1 -1 0 0 0 0 -1 0 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 1 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],[0,0,-1,0,0,0,0,-1,-1,0,0,0,0,-1,0,0],[-1,1,0,0,-1,0,0,0,0,0,-1,1,0,0,-1,0],[0,0,-1,1,0,0,0,1,-1,1,0,0,0,1,0,0] >;

C2×S32 in GAP, Magma, Sage, TeX

C_2\times S_3^2
% in TeX

G:=Group("C2xS3^2");
// GroupNames label

G:=SmallGroup(72,46);
// by ID

G=gap.SmallGroup(72,46);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,168,1204]);
// Polycyclic

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

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