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G = S3xC2xC4  order 48 = 24·3

Direct product of C2xC4 and S3

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

Aliases: S3xC2xC4, C12:3C22, C6.2C23, C22.9D6, D6.4C22, Dic3:3C22, C4o(C4xS3), C6:1(C2xC4), (C2xC12):5C2, C3:1(C22xC4), (C2xC4)oDic3, C4o(C2xDic3), (C2xDic3):5C2, C2.1(C22xS3), (C2xC6).9C22, (C22xS3).2C2, (C2xC4)o(C2xDic3), SmallGroup(48,35)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — S3xC2xC4
Chief series
C1 — C3 — C6 — D6 — C22xS3 — S3xC2xC4
Lower central
C3 — S3xC2xC4
Upper central
C1 — C2xC4

Generators and relations for S3xC2xC4
 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, C2xC4, C2xC4, C23, Dic3, C12, D6, C2xC6, C22xC4, C4xS3, C2xDic3, C2xC12, C22xS3, S3xC2xC4
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C4xS3, C22xS3, S3xC2xC4

Character table of S3xC2xC4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H6A6B6C12A12B12C12D
 size 111133332111133332222222
ρ1111111111111111111111111    trivial
ρ21111-1-1-1-111111-1-1-1-11111111    linear of order 2
ρ3111111111-1-1-1-1-1-1-1-1111-1-1-1-1    linear of order 2
ρ41111-1-1-1-11-1-1-1-11111111-1-1-1-1    linear of order 2
ρ511-1-1-111-11-11-11-111-11-1-111-1-1    linear of order 2
ρ611-1-11-1-111-11-111-1-111-1-111-1-1    linear of order 2
ρ711-1-1-111-111-11-11-1-111-1-1-1-111    linear of order 2
ρ811-1-11-1-1111-11-1-111-11-1-1-1-111    linear of order 2
ρ91-1-1111-1-11-iii-iii-i-i-11-1i-ii-i    linear of order 4
ρ101-1-11-1-1111i-i-iiii-i-i-11-1-ii-ii    linear of order 4
ρ111-11-1-11-111ii-i-i-ii-ii-1-11i-i-ii    linear of order 4
ρ121-11-11-11-11-i-iii-ii-ii-1-11-iii-i    linear of order 4
ρ131-1-11-1-1111-iii-i-i-iii-11-1i-ii-i    linear of order 4
ρ141-1-1111-1-11i-i-ii-i-iii-11-1-ii-ii    linear of order 4
ρ151-11-11-11-11ii-i-ii-ii-i-1-11i-i-ii    linear of order 4
ρ161-11-1-11-111-i-iiii-ii-i-1-11-iii-i    linear of order 4
ρ1722220000-1-2-2-2-20000-1-1-11111    orthogonal lifted from D6
ρ1822-2-20000-1-22-220000-111-1-111    orthogonal lifted from D6
ρ1922-2-20000-12-22-20000-11111-1-1    orthogonal lifted from D6
ρ2022220000-122220000-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ212-22-20000-12i2i-2i-2i000011-1-iii-i    complex lifted from C4xS3
ρ222-2-220000-12i-2i-2i2i00001-11i-ii-i    complex lifted from C4xS3
ρ232-2-220000-1-2i2i2i-2i00001-11-ii-ii    complex lifted from C4xS3
ρ242-22-20000-1-2i-2i2i2i000011-1i-i-ii    complex lifted from C4xS3

Permutation representations of S3xC2xC4
►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);
 

S3xC2xC4 is a maximal subgroup of
 D6:C8  C42:2S3  Dic3:4D4  C23.9D6  Dic3:D4  C4:C4:7S3  Dic3:5D4  D6.D4  C12:D4  D6:Q8  C4.D12  D6:3D4  D6:3Q8
S3xC2xC4 is a maximal quotient of
 C42:2S3  C23.16D6  Dic3:4D4  Dic6:C4  C4:C4:7S3  Dic3:5D4  C8oD12  D12.C4

Matrix representation of S3xC2xC4 ►in GL3(F13) generated by

1200
010
001
,
800
050
005
,
100
01212
010
,
1200
010
01212
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] >;
 

S3xC2xC4 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
 

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Character table of S3xC2xC4 in TeX

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