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G = S3×C8order 48 = 24·3

Direct product of C8 and S3

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

Aliases: S3×C8, C244C2, C8Dic3, D6.2C4, C4.12D6, Dic3.2C4, C12.12C22, C8(C3⋊C8), C3⋊C86C2, C31(C2×C8), C2.1(C4×S3), C6.1(C2×C4), (C4×S3).3C2, SmallGroup(48,4)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C8
C1C3C6C12C4×S3 — S3×C8
C3 — S3×C8
C1C8

Generators and relations for S3×C8
 G = < a,b,c | a8=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
3C22
3C4
3C2×C4
3C8
3C2×C8

Character table of S3×C8

 class 12A2B2C34A4B4C4D68A8B8C8D8E8F8G8H12A12B24A24B24C24D
 size 113321133211113333222222
ρ1111111111111111111111111    trivial
ρ211-1-1111-1-11-1-1-1-1111111-1-1-1-1    linear of order 2
ρ31111111111-1-1-1-1-1-1-1-111-1-1-1-1    linear of order 2
ρ411-1-1111-1-111111-1-1-1-1111111    linear of order 2
ρ511-1-11-1-1111-ii-iii-i-ii-1-1-i-iii    linear of order 4
ρ611111-1-1-1-11i-ii-ii-i-ii-1-1ii-i-i    linear of order 4
ρ711-1-11-1-1111i-ii-i-iii-i-1-1ii-i-i    linear of order 4
ρ811111-1-1-1-11-ii-ii-iii-i-1-1-i-iii    linear of order 4
ρ91-1-111i-ii-i-1ζ8ζ83ζ85ζ87ζ85ζ83ζ87ζ8-iiζ8ζ85ζ83ζ87    linear of order 8
ρ101-11-11-iii-i-1ζ87ζ85ζ83ζ8ζ87ζ8ζ85ζ83i-iζ87ζ83ζ85ζ8    linear of order 8
ρ111-1-111-ii-ii-1ζ83ζ8ζ87ζ85ζ87ζ8ζ85ζ83i-iζ83ζ87ζ8ζ85    linear of order 8
ρ121-11-11i-i-ii-1ζ85ζ87ζ8ζ83ζ85ζ83ζ87ζ8-iiζ85ζ8ζ87ζ83    linear of order 8
ρ131-11-11-iii-i-1ζ83ζ8ζ87ζ85ζ83ζ85ζ8ζ87i-iζ83ζ87ζ8ζ85    linear of order 8
ρ141-1-111-ii-ii-1ζ87ζ85ζ83ζ8ζ83ζ85ζ8ζ87i-iζ87ζ83ζ85ζ8    linear of order 8
ρ151-11-11i-i-ii-1ζ8ζ83ζ85ζ87ζ8ζ87ζ83ζ85-iiζ8ζ85ζ83ζ87    linear of order 8
ρ161-1-111i-ii-i-1ζ85ζ87ζ8ζ83ζ8ζ87ζ83ζ85-iiζ85ζ8ζ87ζ83    linear of order 8
ρ172200-12200-122220000-1-1-1-1-1-1    orthogonal lifted from S3
ρ182200-12200-1-2-2-2-20000-1-11111    orthogonal lifted from D6
ρ192200-1-2-200-1-2i2i-2i2i000011ii-i-i    complex lifted from C4×S3
ρ202200-1-2-200-12i-2i2i-2i000011-i-iii    complex lifted from C4×S3
ρ212-200-1-2i2i00187858380000-iiζ83ζ87ζ8ζ85    complex faithful
ρ222-200-12i-2i00188385870000i-iζ85ζ8ζ87ζ83    complex faithful
ρ232-200-1-2i2i00183887850000-iiζ87ζ83ζ85ζ8    complex faithful
ρ242-200-12i-2i00185878830000i-iζ8ζ85ζ83ζ87    complex faithful

Permutation representations of S3×C8
On 24 points - transitive group 24T32
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)
(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)

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), (9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20)>;

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), (9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20) );

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)], [(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)])

G:=TransitiveGroup(24,32);

S3×C8 is a maximal subgroup of
D6.C8  C8○D12  D12.C4  D83S3  Q8.7D6  D24⋊C2  C12.29D6  CU2(𝔽3)  D152C8  D15⋊C8  D21⋊C8  C335(C2×C8)  A5⋊C8  C8.A5
S3×C8 is a maximal quotient of
D6.C8  Dic3⋊C8  D6⋊C8  C12.29D6  D152C8  D15⋊C8  D21⋊C8  C335(C2×C8)

Matrix representation of S3×C8 in GL2(𝔽17) generated by

80
08
,
013
1316
,
164
01
G:=sub<GL(2,GF(17))| [8,0,0,8],[0,13,13,16],[16,0,4,1] >;

S3×C8 in GAP, Magma, Sage, TeX

S_3\times C_8
% in TeX

G:=Group("S3xC8");
// GroupNames label

G:=SmallGroup(48,4);
// by ID

G=gap.SmallGroup(48,4);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,26,42,804]);
// Polycyclic

G:=Group<a,b,c|a^8=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

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

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