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G = S3×C2×C4  order 48 = 24·3

Direct product of C2×C4 and S3

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

Aliases: S3×C2×C4, C12⋊3C22, C6.2C23, C22.9D6, D6.4C22, Dic3⋊3C22, C4○(C4×S3), C6⋊1(C2×C4), (C2×C12)⋊5C2, C3⋊1(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 [×2], C2 [×4], C3, C4 [×2], C4 [×2], C22, C22 [×6], S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×5], C23, Dic3 [×2], C12 [×2], D6 [×6], C2×C6, C22×C4, C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, S3×C2×C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, C4×S3 [×2], C22×S3, S3×C2×C4

Character table of S3×C2×C4

 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 C4×S3
ρ222-2-220000-12i-2i-2i2i00001-11i-ii-i    complex lifted from C4×S3
ρ232-2-220000-1-2i2i2i-2i00001-11-ii-ii    complex lifted from C4×S3
ρ242-22-20000-1-2i-2i2i2i000011-1i-i-ii    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);
 

Matrix representation of S3×C2×C4 ►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] >;
 

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
 

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