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G = C22×S3order 24 = 23·3

Direct product of C22 and S3

direct product, metabelian, supersoluble, monomial, A-group, rational, 2-hyperelementary

Aliases: C22×S3, C3⋊C23, C6⋊C22, (C2×C6)⋊3C2, SmallGroup(24,14)

Series: Derived Chief Lower central Upper central

C1C3 — C22×S3
C1C3S3D6 — C22×S3
C3 — C22×S3
C1C22

Generators and relations for C22×S3
 G = < a,b,c,d | a2=b2=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

3C2
3C2
3C2
3C2
3C22
3C22
3C22
3C22
3C22
3C22
3C23

Character table of C22×S3

 class 12A2B2C2D2E2F2G36A6B6C
 size 111133332222
ρ1111111111111    trivial
ρ21-1-111-1-1111-1-1    linear of order 2
ρ311-1-111-1-11-11-1    linear of order 2
ρ41-11-11-11-11-1-11    linear of order 2
ρ511-1-1-1-1111-11-1    linear of order 2
ρ61-11-1-11-111-1-11    linear of order 2
ρ71111-1-1-1-11111    linear of order 2
ρ81-1-11-111-111-1-1    linear of order 2
ρ922220000-1-1-1-1    orthogonal lifted from S3
ρ102-22-20000-111-1    orthogonal lifted from D6
ρ112-2-220000-1-111    orthogonal lifted from D6
ρ1222-2-20000-11-11    orthogonal lifted from D6

Permutation representations of C22×S3
On 12 points - transitive group 12T10
Generators in S12
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)
(1 7)(2 9)(3 8)(4 10)(5 12)(6 11)

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

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

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

G:=TransitiveGroup(12,10);

Regular action on 24 points - transitive group 24T11
Generators in S24
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 4)(2 5)(3 6)(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 17)(2 16)(3 18)(4 14)(5 13)(6 15)(7 23)(8 22)(9 24)(10 20)(11 19)(12 21)

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

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

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

G:=TransitiveGroup(24,11);

Polynomial with Galois group C22×S3 over ℚ
actionf(x)Disc(f)
12T10x12-4x6+1224·318

Matrix representation of C22×S3 in GL3(ℤ) generated by

100
0-10
00-1
,
-100
010
001
,
100
00-1
01-1
,
-100
00-1
0-10
G:=sub<GL(3,Integers())| [1,0,0,0,-1,0,0,0,-1],[-1,0,0,0,1,0,0,0,1],[1,0,0,0,0,1,0,-1,-1],[-1,0,0,0,0,-1,0,-1,0] >;

C22×S3 in GAP, Magma, Sage, TeX

C_2^2\times S_3
% in TeX

G:=Group("C2^2xS3");
// GroupNames label

G:=SmallGroup(24,14);
// by ID

G=gap.SmallGroup(24,14);
# by ID

G:=PCGroup([4,-2,-2,-2,-3,259]);
// Polycyclic

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

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