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G = C8⋊S3  order 48 = 24·3

3rd semidirect product of C8 and S3 acting via S3/C3=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C8⋊3S3, D6.C4, C24⋊5C2, C4.13D6, Dic3.C4, C3⋊1M4(2), C12.13C22, C3⋊C8⋊4C2, C2.3(C4×S3), C6.2(C2×C4), (C4×S3).2C2, SmallGroup(48,5)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C8⋊S3
Chief series
C1 — C3 — C6 — C12 — C4×S3 — C8⋊S3
Lower central
C3 — C6 — C8⋊S3
Upper central
C1 — C4 — C8

Generators and relations for C8⋊S3
 G = < a,b,c | a8=b3=c2=1, ab=ba, cac=a5, cbc=b-1 >

C1
C2
6C2
C3
C4
3C22
3C4
C6
2S3
C8
3C2×C4
3C8
D6
Dic3
C12
3M4(2)
C4×S3
C3⋊C8
C24
C8⋊S3

Character table of C8⋊S3

 class 12A2B34A4B4C68A8B8C8D12A12B24A24B24C24D
 size 116211622266222222
ρ1111111111111111111    trivial
ρ211-1111-1111-1-1111111    linear of order 2
ρ311111111-1-1-1-111-1-1-1-1    linear of order 2
ρ411-1111-11-1-11111-1-1-1-1    linear of order 2
ρ511-11-1-111-ii-ii-1-1-ii-ii    linear of order 4
ρ61111-1-1-11-iii-i-1-1-ii-ii    linear of order 4
ρ711-11-1-111i-ii-i-1-1i-ii-i    linear of order 4
ρ81111-1-1-11i-i-ii-1-1i-ii-i    linear of order 4
ρ9220-1220-12200-1-1-1-1-1-1    orthogonal lifted from S3
ρ10220-1220-1-2-200-1-11111    orthogonal lifted from D6
ρ112-2022i-2i0-200002i-2i0000    complex lifted from M4(2)
ρ12220-1-2-20-12i-2i0011-ii-ii    complex lifted from C4×S3
ρ13220-1-2-20-1-2i2i0011i-ii-i    complex lifted from C4×S3
ρ142-202-2i2i0-20000-2i2i0000    complex lifted from M4(2)
ρ152-20-1-2i2i010000i-i2ζ85ζ3+ζ852ζ87ζ3+ζ872ζ8ζ3+ζ82ζ83ζ3+ζ83    complex faithful
ρ162-20-12i-2i010000-ii2ζ83ζ3+ζ832ζ8ζ3+ζ82ζ87ζ3+ζ872ζ85ζ3+ζ85    complex faithful
ρ172-20-12i-2i010000-ii2ζ87ζ3+ζ872ζ85ζ3+ζ852ζ83ζ3+ζ832ζ8ζ3+ζ8    complex faithful
ρ182-20-1-2i2i010000i-i2ζ8ζ3+ζ82ζ83ζ3+ζ832ζ85ζ3+ζ852ζ87ζ3+ζ87    complex faithful

Permutation representations of C8⋊S3
►On 24 points - transitive group 24T31
Generators in S24
(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 9 21)(2 10 22)(3 11 23)(4 12 24)(5 13 17)(6 14 18)(7 15 19)(8 16 20)

(2 6)(4 8)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)
 
G:=sub<Sym(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), (1,9,21)(2,10,22)(3,11,23)(4,12,24)(5,13,17)(6,14,18)(7,15,19)(8,16,20), (2,6)(4,8)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,9,21)(2,10,22)(3,11,23)(4,12,24)(5,13,17)(6,14,18)(7,15,19)(8,16,20), (2,6)(4,8)(9,21)(10,18)(11,23)(12,20)(13,17)(14,22)(15,19)(16,24) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,9,21),(2,10,22),(3,11,23),(4,12,24),(5,13,17),(6,14,18),(7,15,19),(8,16,20)], [(2,6),(4,8),(9,21),(10,18),(11,23),(12,20),(13,17),(14,22),(15,19),(16,24)]])
 
G:=TransitiveGroup(24,31);
 

C8⋊S3 is a maximal subgroup of
 C8○D12  S3×M4(2)  D12.C4  D8⋊S3  Q8⋊3D6  D4.D6  Q16⋊S3  C8⋊D9  D6.Dic3  C12.31D6  C24⋊S3  C8⋊S4  C8.5S4  D6.Dic5  D30.5C4  C40⋊S3  D6.F5  Dic3.F5  D6.Dic7  D42.C4  C56⋊S3  C33⋊M4(2)  C33⋊2M4(2)  GL2(𝔽5)
C8⋊S3 is a maximal quotient of
 Dic3⋊C8  C24⋊C4  D6⋊C8  C8⋊D9  D6.Dic3  C12.31D6  C24⋊S3  C8⋊S4  D6.Dic5  D30.5C4  C40⋊S3  D6.F5  Dic3.F5  D6.Dic7  D42.C4  C56⋊S3  C33⋊M4(2)  C33⋊2M4(2)

Matrix representation of C8⋊S3 ►in GL2(𝔽5) generated by

21
43
,
41
40
,
14
04
G:=sub<GL(2,GF(5))| [2,4,1,3],[4,4,1,0],[1,0,4,4] >;
 

C8⋊S3 in GAP, Magma, Sage, TeX 

C_8\rtimes S_3
 
% in TeX
 
G:=Group("C8:S3");
 
// GroupNames label
 
G:=SmallGroup(48,5);
 
// by ID
 
G=gap.SmallGroup(48,5);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-2,-3,101,26,42,804]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^3=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C8⋊S3 in TeX
Character table of C8⋊S3 in TeX

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