Copied to
clipboard

## G = S3×C2×C4order 48 = 24·3

### Direct product of C2×C4 and S3

Aliases: S3×C2×C4, C123C22, C6.2C23, C22.9D6, D6.4C22, Dic33C22, C4(C4×S3), C61(C2×C4), (C2×C12)⋊5C2, C31(C22×C4), (C2×C4)Dic3, C4(C2×Dic3), (C2×Dic3)⋊5C2, C2.1(C22×S3), (C2×C6).9C22, (C22×S3).2C2, (C2×C4)(C2×Dic3), SmallGroup(48,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C4
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C2×C4
 Lower central C3 — S3×C2×C4
 Upper central C1 — C2×C4

Generators and relations for S3×C2×C4
G = < a,b,c,d | a2=b4=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 92 in 54 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, C4×S3, C2×Dic3, C2×C12, C22×S3, S3×C2×C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, S3×C2×C4

Character table of S3×C2×C4

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 12A 12B 12C 12D size 1 1 1 1 3 3 3 3 2 1 1 1 1 3 3 3 3 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 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 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 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 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 -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 -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 -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 -1 -1 -1 -1 1 1 linear of order 2 ρ9 1 -1 -1 1 1 1 -1 -1 1 -i i i -i i i -i -i -1 1 -1 i -i i -i linear of order 4 ρ10 1 -1 -1 1 -1 -1 1 1 1 i -i -i i i i -i -i -1 1 -1 -i i -i i linear of order 4 ρ11 1 -1 1 -1 -1 1 -1 1 1 i i -i -i -i i -i i -1 -1 1 i -i -i i linear of order 4 ρ12 1 -1 1 -1 1 -1 1 -1 1 -i -i i i -i i -i i -1 -1 1 -i i i -i linear of order 4 ρ13 1 -1 -1 1 -1 -1 1 1 1 -i i i -i -i -i i i -1 1 -1 i -i i -i linear of order 4 ρ14 1 -1 -1 1 1 1 -1 -1 1 i -i -i i -i -i i i -1 1 -1 -i i -i i linear of order 4 ρ15 1 -1 1 -1 1 -1 1 -1 1 i i -i -i i -i i -i -1 -1 1 i -i -i i linear of order 4 ρ16 1 -1 1 -1 -1 1 -1 1 1 -i -i i i i -i i -i -1 -1 1 -i i i -i linear of order 4 ρ17 2 2 2 2 0 0 0 0 -1 -2 -2 -2 -2 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ18 2 2 -2 -2 0 0 0 0 -1 -2 2 -2 2 0 0 0 0 -1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ19 2 2 -2 -2 0 0 0 0 -1 2 -2 2 -2 0 0 0 0 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ20 2 2 2 2 0 0 0 0 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ21 2 -2 2 -2 0 0 0 0 -1 2i 2i -2i -2i 0 0 0 0 1 1 -1 -i i i -i complex lifted from C4×S3 ρ22 2 -2 -2 2 0 0 0 0 -1 2i -2i -2i 2i 0 0 0 0 1 -1 1 i -i i -i complex lifted from C4×S3 ρ23 2 -2 -2 2 0 0 0 0 -1 -2i 2i 2i -2i 0 0 0 0 1 -1 1 -i i -i i complex lifted from C4×S3 ρ24 2 -2 2 -2 0 0 0 0 -1 -2i -2i 2i 2i 0 0 0 0 1 1 -1 i -i -i i complex lifted from C4×S3

Permutation representations of S3×C2×C4
On 24 points - transitive group 24T27
Generators in S24
(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 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 20)(14 17)(15 18)(16 19)(21 23)(22 24)

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

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

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

G:=TransitiveGroup(24,27);

S3×C2×C4 is a maximal subgroup of
D6⋊C8  C422S3  Dic34D4  C23.9D6  Dic3⋊D4  C4⋊C47S3  Dic35D4  D6.D4  C12⋊D4  D6⋊Q8  C4.D12  D63D4  D63Q8
S3×C2×C4 is a maximal quotient of
C422S3  C23.16D6  Dic34D4  Dic6⋊C4  C4⋊C47S3  Dic35D4  C8○D12  D12.C4

Matrix representation of S3×C2×C4 in GL3(𝔽13) generated by

 12 0 0 0 1 0 0 0 1
,
 8 0 0 0 5 0 0 0 5
,
 1 0 0 0 12 12 0 1 0
,
 12 0 0 0 1 0 0 12 12
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,1],[8,0,0,0,5,0,0,0,5],[1,0,0,0,12,1,0,12,0],[12,0,0,0,1,12,0,0,12] >;

S3×C2×C4 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_4
% in TeX

G:=Group("S3xC2xC4");
// GroupNames label

G:=SmallGroup(48,35);
// by ID

G=gap.SmallGroup(48,35);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,42,804]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=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

Export

׿
×
𝔽