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G = S3×C23  order 48 = 24·3

Direct product of C23 and S3

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

Aliases: S3×C23, C3⋊C24, C6⋊C23, (C22×C6)⋊3C2, (C2×C6)⋊4C22, SmallGroup(48,51)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — S3×C23
Chief series
C1 — C3 — S3 — D6 — C22×S3 — S3×C23
Lower central
C3 — S3×C23
Upper central
C1 — C23

Generators and relations for S3×C23
 G = < a,b,c,d,e | a2=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 236 in 134 conjugacy classes, 83 normal (5 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, C23, D6, C2×C6, C24, C22×S3, C22×C6, S3×C23
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, S3×C23

Character table of S3×C23

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O36A6B6C6D6E6F6G
 size 111111113333333322222222
ρ1111111111111111111111111    trivial
ρ2111-11-1-1-11-11-11-11-111-1-11-1-11    linear of order 2
ρ311-1-1-1-111-1-1-11111-111-11-1-11-1    linear of order 2
ρ411-11-11-1-1-11-1-11-111111-1-11-1-1    linear of order 2
ρ511-1-1-1-111111-1-1-1-1111-11-1-11-1    linear of order 2
ρ611-11-11-1-11-111-11-1-1111-1-11-1-1    linear of order 2
ρ711111111-1-1-1-1-1-1-1-111111111    linear of order 2
ρ8111-11-1-1-1-11-11-11-1111-1-11-1-11    linear of order 2
ρ91-1-111-1-11-1-111-1-1111-1-1-1111-1    linear of order 2
ρ101-1-1-1111-1-111-1-111-11-1111-1-1-1    linear of order 2
ρ111-11-1-11-1111-11-1-11-11-11-1-1-111    linear of order 2
ρ121-111-1-11-11-1-1-1-11111-1-11-11-11    linear of order 2
ρ131-11-1-11-11-1-11-111-111-11-1-1-111    linear of order 2
ρ141-111-1-11-1-11111-1-1-11-1-11-11-11    linear of order 2
ρ151-1-111-1-1111-1-111-1-11-1-1-1111-1    linear of order 2
ρ161-1-1-1111-11-1-111-1-111-1111-1-1-1    linear of order 2
ρ172-222-2-22-200000000-111-11-11-1    orthogonal lifted from D6
ρ1822-22-22-2-200000000-1-1-111-111    orthogonal lifted from D6
ρ192-22-2-22-2200000000-11-1111-1-1    orthogonal lifted from D6
ρ2022-2-2-2-22200000000-1-11-111-11    orthogonal lifted from D6
ρ21222-22-2-2-200000000-1-111-111-1    orthogonal lifted from D6
ρ222-2-2-2222-200000000-11-1-1-1111    orthogonal lifted from D6
ρ232222222200000000-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ242-2-222-2-2200000000-1111-1-1-11    orthogonal lifted from D6

Permutation representations of S3×C23
►On 24 points - transitive group 24T30
Generators in S24
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 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 16)(2 18)(3 17)(4 13)(5 15)(6 14)(7 22)(8 24)(9 23)(10 19)(11 21)(12 20)
 
G:=sub<Sym(24)| (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (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), (1,16)(2,18)(3,17)(4,13)(5,15)(6,14)(7,22)(8,24)(9,23)(10,19)(11,21)(12,20)>;
 
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (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), (1,16)(2,18)(3,17)(4,13)(5,15)(6,14)(7,22)(8,24)(9,23)(10,19)(11,21)(12,20) );
 
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(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)], [(1,16),(2,18),(3,17),(4,13),(5,15),(6,14),(7,22),(8,24),(9,23),(10,19),(11,21),(12,20)]])
 
G:=TransitiveGroup(24,30);
 

S3×C23 is a maximal subgroup of   D6⋊D4  C23⋊2D6
S3×C23 is a maximal quotient of   D4⋊6D6  Q8.15D6  D4○D12  Q8○D12

Matrix representation of S3×C23 ►in GL4(ℤ) generated by

1000
0-100
0010
0001
,
1000
0-100
00-10
000-1
,
-1000
0100
0010
0001
,
1000
0100
00-1-1
0010
,
-1000
0100
0010
00-1-1
G:=sub<GL(4,Integers())| [1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[-1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,-1,1,0,0,-1,0],[-1,0,0,0,0,1,0,0,0,0,1,-1,0,0,0,-1] >;
 

S3×C23 in GAP, Magma, Sage, TeX 

S_3\times C_2^3
 
% in TeX
 
G:=Group("S3xC2^3");
 
// GroupNames label
 
G:=SmallGroup(48,51);
 
// by ID
 
G=gap.SmallGroup(48,51);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-2,-3,804]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,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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Character table of S3×C23 in TeX

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