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G = S3×D12order 144 = 24·32

Direct product of S3 and D12

direct product, metabelian, supersoluble, monomial

Aliases: S3×D12, D61D6, C121D6, Dic33D6, C41S32, C31(S3×D4), (C4×S3)⋊3S3, (C3×S3)⋊1D4, C31(C2×D12), (S3×C12)⋊4C2, C322(C2×D4), (C3×D12)⋊6C2, C12⋊S35C2, C3⋊D121C2, (S3×C6)⋊1C22, (C3×C12)⋊1C22, (C3×C6).8C23, C6.8(C22×S3), (C3×Dic3)⋊3C22, (C2×S32)⋊1C2, C2.10(C2×S32), (C2×C3⋊S3)⋊1C22, SmallGroup(144,144)

Series: Derived Chief Lower central Upper central

C1C3×C6 — S3×D12
C1C3C32C3×C6S3×C6C2×S32 — S3×D12
C32C3×C6 — S3×D12
C1C2C4

Generators and relations for S3×D12
 G = < a,b,c,d | a3=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 476 in 116 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4, C22 [×9], S3 [×2], S3 [×8], C6 [×2], C6 [×5], C2×C4, D4 [×4], C23 [×2], C32, Dic3, C12 [×2], C12 [×2], D6, D6 [×2], D6 [×14], C2×C6 [×3], C2×D4, C3×S3 [×2], C3×S3 [×2], C3⋊S3 [×2], C3×C6, C4×S3, D12, D12 [×5], C3⋊D4 [×2], C2×C12, C3×D4, C22×S3 [×4], C3×Dic3, C3×C12, S32 [×4], S3×C6, S3×C6 [×2], C2×C3⋊S3 [×2], C2×D12, S3×D4, C3⋊D12 [×2], S3×C12, C3×D12, C12⋊S3, C2×S32 [×2], S3×D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C22×S3 [×2], S32, C2×D12, S3×D4, C2×S32, S3×D12

Character table of S3×D12

 class 12A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D6E6F6G12A12B12C12D12E12F12G
 size 1133661818224262246612122244466
ρ1111111111111111111111111111    trivial
ρ21111-111-1111-1-111111-11-1-1-1-1-1-1-1    linear of order 2
ρ311-1-111-1-11111-1111-1-11111111-1-1    linear of order 2
ρ411-1-1-11-11111-11111-1-1-11-1-1-1-1-111    linear of order 2
ρ51111-1-1-1-11111111111-1-11111111    linear of order 2
ρ611111-1-11111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ711-1-1-1-1111111-1111-1-1-1-111111-1-1    linear of order 2
ρ811-1-11-11-1111-11111-1-11-1-1-1-1-1-111    linear of order 2
ρ922-2-200002-1-1-222-1-11100111-21-1-1    orthogonal lifted from D6
ρ102-2-22000022200-2-2-2-22000000000    orthogonal lifted from D4
ρ11222200002-1-1-2-22-1-1-1-100111-2111    orthogonal lifted from D6
ρ122-22-2000022200-2-2-22-2000000000    orthogonal lifted from D4
ρ1322002-200-12-1-20-12-100-11-2-211100    orthogonal lifted from D6
ρ14222200002-1-1222-1-1-1-100-1-1-12-1-1-1    orthogonal lifted from S3
ρ1522002200-12-120-12-100-1-122-1-1-100    orthogonal lifted from S3
ρ1622-2-200002-1-12-22-1-11100-1-1-12-111    orthogonal lifted from D6
ρ172200-2-200-12-120-12-1001122-1-1-100    orthogonal lifted from D6
ρ182200-2200-12-1-20-12-1001-1-2-211100    orthogonal lifted from D6
ρ192-2-2200002-1-100-2111-1003-330-3-33    orthogonal lifted from D12
ρ202-22-200002-1-100-211-11003-330-33-3    orthogonal lifted from D12
ρ212-2-2200002-1-100-2111-100-33-3033-3    orthogonal lifted from D12
ρ222-22-200002-1-100-211-1100-33-303-33    orthogonal lifted from D12
ρ234-4000000-24-2002-4200000000000    orthogonal lifted from S3×D4
ρ2444000000-2-2140-2-210000-2-21-2100    orthogonal lifted from S32
ρ2544000000-2-21-40-2-21000022-12-100    orthogonal lifted from C2×S32
ρ264-4000000-2-210022-1000023-23-30300    orthogonal faithful
ρ274-4000000-2-210022-10000-232330-300    orthogonal faithful

Permutation representations of S3×D12
On 24 points - transitive group 24T229
Generators in S24
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(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)
(1 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(24,229);

S3×D12 is a maximal subgroup of
C241D6  D24⋊S3  Dic63D6  D126D6  D1224D6  D1227D6  S32×D4  D1213D6  D1216D6  C3⋊S3⋊D12  C12.86S32  C3⋊S34D12  C123S32
S3×D12 is a maximal quotient of
C241D6  D24⋊S3  C24.3D6  Dic12⋊S3  D6.1D12  D247S3  D6.3D12  Dic3.D12  Dic34D12  Dic3⋊D12  D6.D12  D6.9D12  D62Dic6  Dic35D12  C62.65C23  D6⋊D12  D62D12  C127D12  Dic33D12  C12⋊D12  C123Dic6  D64D12  D65D12  C3⋊S3⋊D12  C3⋊S34D12  C123S32

Matrix representation of S3×D12 in GL6(ℤ)

100000
010000
000100
00-1-100
000010
000001
,
-100000
0-10000
00-1000
001100
000010
000001
,
120000
-1-10000
001000
000100
0000-11
0000-10
,
100000
-1-10000
00-1000
000-100
0000-10
0000-11

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,-1,0,0,0,0,2,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1] >;

S3×D12 in GAP, Magma, Sage, TeX

S_3\times D_{12}
% in TeX

G:=Group("S3xD12");
// GroupNames label

G:=SmallGroup(144,144);
// by ID

G=gap.SmallGroup(144,144);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,116,50,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^12=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

Export

Character table of S3×D12 in TeX

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