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G = C8⋊S3order 48 = 24·3

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

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C83S3, D6.C4, C245C2, C4.13D6, Dic3.C4, C31M4(2), C12.13C22, C3⋊C84C2, C2.3(C4×S3), C6.2(C2×C4), (C4×S3).2C2, SmallGroup(48,5)

Series: Derived Chief Lower central Upper central

C1C6 — C8⋊S3
C1C3C6C12C4×S3 — C8⋊S3
C3C6 — C8⋊S3
C1C4C8

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

6C2
3C22
3C4
2S3
3C2×C4
3C8
3M4(2)

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)
ρ152-20-1-2i2i010000ζ2ζ285ζ38587ζ3878ζ3883ζ383    complex faithful
ρ162-20-12i-2i010000ζ2ζ283ζ3838ζ3887ζ38785ζ385    complex faithful
ρ172-20-12i-2i010000ζ2ζ287ζ38785ζ38583ζ3838ζ38    complex faithful
ρ182-20-1-2i2i010000ζ2ζ28ζ3883ζ38385ζ38587ζ387    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 15 21)(2 16 22)(3 9 23)(4 10 24)(5 11 17)(6 12 18)(7 13 19)(8 14 20)
(2 6)(4 8)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)

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

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

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

G:=TransitiveGroup(24,31);

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

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