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## G = S3×D4order 48 = 24·3

### Direct product of S3 and D4

Aliases: S3×D4, C41D6, C12⋊C22, D123C2, C222D6, D62C22, C6.5C23, Dic31C22, C32(C2×D4), (C2×C6)⋊C22, (C4×S3)⋊1C2, (C3×D4)⋊2C2, C3⋊D41C2, (C22×S3)⋊2C2, C2.6(C22×S3), Aut(D12), Hol(C12), SmallGroup(48,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×D4
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×D4
 Lower central C3 — C6 — S3×D4
 Upper central C1 — C2 — D4

Generators and relations for S3×D4
G = < a,b,c,d | a3=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 120 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×2], C22 [×7], S3 [×2], S3 [×2], C6, C6 [×2], C2×C4, D4, D4 [×3], C23 [×2], Dic3, C12, D6, D6 [×2], D6 [×4], C2×C6 [×2], C2×D4, C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], S3×D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C22×S3, S3×D4

Character table of S3×D4

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 12 size 1 1 2 2 3 3 6 6 2 2 6 2 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 1 1 1 linear of order 2 ρ3 1 1 1 -1 1 1 1 -1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ4 1 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 1 -1 linear of order 2 ρ5 1 1 -1 1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 linear of order 2 ρ6 1 1 -1 1 1 1 -1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ7 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 1 linear of order 2 ρ8 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 linear of order 2 ρ9 2 2 -2 2 0 0 0 0 -1 -2 0 -1 -1 1 1 orthogonal lifted from D6 ρ10 2 -2 0 0 2 -2 0 0 2 0 0 -2 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 0 0 -1 2 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 -2 0 0 0 0 -1 -2 0 -1 1 -1 1 orthogonal lifted from D6 ρ13 2 2 -2 -2 0 0 0 0 -1 2 0 -1 1 1 -1 orthogonal lifted from D6 ρ14 2 -2 0 0 -2 2 0 0 2 0 0 -2 0 0 0 orthogonal lifted from D4 ρ15 4 -4 0 0 0 0 0 0 -2 0 0 2 0 0 0 orthogonal faithful

Permutation representations of S3×D4
On 12 points - transitive group 12T28
Generators in S12
(1 9 8)(2 10 5)(3 11 6)(4 12 7)
(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 7)(10 12)

G:=sub<Sym(12)| (1,9,8)(2,10,5)(3,11,6)(4,12,7), (5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,7)(10,12)>;

G:=Group( (1,9,8)(2,10,5)(3,11,6)(4,12,7), (5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,7)(10,12) );

G=PermutationGroup([(1,9,8),(2,10,5),(3,11,6),(4,12,7)], [(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(2,4),(5,7),(10,12)])

G:=TransitiveGroup(12,28);

On 24 points - transitive group 24T52
Generators in S24
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 24)(2 21)(3 22)(4 23)(5 15)(6 16)(7 13)(8 14)(9 19)(10 20)(11 17)(12 18)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(5 7)(10 12)(13 15)(18 20)(21 23)

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

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

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

G:=TransitiveGroup(24,52);

On 24 points - transitive group 24T53
Generators in S24
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(1 3)(2 4)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)(17 24)(18 21)(19 22)(20 23)
(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 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,53);

On 24 points - transitive group 24T54
Generators in S24
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(1 6)(2 7)(3 8)(4 5)(9 23)(10 24)(11 21)(12 22)(13 18)(14 19)(15 20)(16 17)
(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 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,54);

Polynomial with Galois group S3×D4 over ℚ
actionf(x)Disc(f)
12T28x12-4x6-5x3-1-318·56·113

Matrix representation of S3×D4 in GL4(ℤ) generated by

 -1 -1 0 0 1 0 0 0 0 0 -1 -1 0 0 1 0
,
 -1 0 0 0 1 1 0 0 0 0 -1 0 0 0 1 1
,
 0 0 -1 0 0 0 0 -1 1 0 0 0 0 1 0 0
,
 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1
G:=sub<GL(4,Integers())| [-1,1,0,0,-1,0,0,0,0,0,-1,1,0,0,-1,0],[-1,1,0,0,0,1,0,0,0,0,-1,1,0,0,0,1],[0,0,1,0,0,0,0,1,-1,0,0,0,0,-1,0,0],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1] >;

S3×D4 in GAP, Magma, Sage, TeX

S_3\times D_4
% in TeX

G:=Group("S3xD4");
// GroupNames label

G:=SmallGroup(48,38);
// by ID

G=gap.SmallGroup(48,38);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,97,804]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^4=d^2=1,b*a*b=a^-1,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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