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G = S3×Dic6order 144 = 24·32

Direct product of S3 and Dic6

direct product, metabelian, supersoluble, monomial

Aliases: S3×Dic6, D6.7D6, C12.20D6, Dic3.5D6, C4.5S32, (C3×S3)⋊Q8, C32(S3×Q8), (C4×S3).1S3, C322(C2×Q8), C31(C2×Dic6), (S3×Dic3).C2, (S3×C12).2C2, (C3×Dic6)⋊4C2, C322Q82C2, (C3×C6).1C23, C6.1(C22×S3), C324Q84C2, (S3×C6).5C22, (C3×C12).16C22, C3⋊Dic3.4C22, (C3×Dic3).1C22, C2.4(C2×S32), SmallGroup(144,137)

Series: Derived Chief Lower central Upper central

C1C3×C6 — S3×Dic6
C1C3C32C3×C6S3×C6S3×Dic3 — S3×Dic6
C32C3×C6 — S3×Dic6
C1C2C4

Generators and relations for S3×Dic6
 G = < a,b,c,d | a3=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 228 in 82 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, C32, Dic3, Dic3, Dic3, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3×C6, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×C12, C3×Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×Dic6, S3×Q8, S3×Dic3, C322Q8, C3×Dic6, S3×C12, C324Q8, S3×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, S32, C2×Dic6, S3×Q8, C2×S32, S3×Dic6

Character table of S3×Dic6

 class 12A2B2C3A3B3C4A4B4C4D4E4F6A6B6C6D6E12A12B12C12D12E12F12G12H12I
 size 1133224266618182246622444661212
ρ1111111111111111111111111111    trivial
ρ211111111-1-11-1-1111111111111-1-1    linear of order 2
ρ31111111-11-1-1-1111111-1-1-1-1-1-1-1-11    linear of order 2
ρ41111111-1-11-11-111111-1-1-1-1-1-1-11-1    linear of order 2
ρ511-1-1111111-1-1-1111-1-111111-1-111    linear of order 2
ρ611-1-11111-1-1-111111-1-111111-1-1-1-1    linear of order 2
ρ711-1-1111-11-111-1111-1-1-1-1-1-1-111-11    linear of order 2
ρ811-1-1111-1-111-11111-1-1-1-1-1-1-1111-1    linear of order 2
ρ92200-12-12220002-1-10022-1-1-100-1-1    orthogonal lifted from S3
ρ1022222-1-1-200-200-12-1-1-111-2111100    orthogonal lifted from D6
ρ112200-12-12-2-20002-1-10022-1-1-10011    orthogonal lifted from D6
ρ1222-2-22-1-1200-200-12-111-1-12-1-11100    orthogonal lifted from D6
ρ132200-12-1-22-20002-1-100-2-2111001-1    orthogonal lifted from D6
ρ1422222-1-1200200-12-1-1-1-1-12-1-1-1-100    orthogonal lifted from S3
ρ1522-2-22-1-1-200200-12-11111-211-1-100    orthogonal lifted from D6
ρ162200-12-1-2-220002-1-100-2-211100-11    orthogonal lifted from D6
ρ172-2-22222000000-2-2-2-22000000000    symplectic lifted from Q8, Schur index 2
ρ182-22-2222000000-2-2-22-2000000000    symplectic lifted from Q8, Schur index 2
ρ192-2-222-1-10000001-211-13-30-33-3300    symplectic lifted from Dic6, Schur index 2
ρ202-2-222-1-10000001-211-1-3303-33-300    symplectic lifted from Dic6, Schur index 2
ρ212-22-22-1-10000001-21-113-30-333-300    symplectic lifted from Dic6, Schur index 2
ρ222-22-22-1-10000001-21-11-3303-3-3300    symplectic lifted from Dic6, Schur index 2
ρ234400-2-21-400000-2-2100222-1-10000    orthogonal lifted from C2×S32
ρ244400-2-21400000-2-2100-2-2-2110000    orthogonal lifted from S32
ρ254-400-24-2000000-42200000000000    symplectic lifted from S3×Q8, Schur index 2
ρ264-400-2-2100000022-100-23230-330000    symplectic faithful, Schur index 2
ρ274-400-2-2100000022-10023-2303-30000    symplectic faithful, Schur index 2

Smallest permutation representation of S3×Dic6
On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 14 7 20)(2 13 8 19)(3 24 9 18)(4 23 10 17)(5 22 11 16)(6 21 12 15)(25 43 31 37)(26 42 32 48)(27 41 33 47)(28 40 34 46)(29 39 35 45)(30 38 36 44)

G:=sub<Sym(48)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,14,7,20)(2,13,8,19)(3,24,9,18)(4,23,10,17)(5,22,11,16)(6,21,12,15)(25,43,31,37)(26,42,32,48)(27,41,33,47)(28,40,34,46)(29,39,35,45)(30,38,36,44)>;

G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,14,7,20)(2,13,8,19)(3,24,9,18)(4,23,10,17)(5,22,11,16)(6,21,12,15)(25,43,31,37)(26,42,32,48)(27,41,33,47)(28,40,34,46)(29,39,35,45)(30,38,36,44) );

G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(1,34),(2,35),(3,36),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,45),(14,46),(15,47),(16,48),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,14,7,20),(2,13,8,19),(3,24,9,18),(4,23,10,17),(5,22,11,16),(6,21,12,15),(25,43,31,37),(26,42,32,48),(27,41,33,47),(28,40,34,46),(29,39,35,45),(30,38,36,44)]])

S3×Dic6 is a maximal subgroup of
C24.3D6  Dic12⋊S3  Dic6.19D6  D12.11D6  D12.33D6  D12.34D6  Dic6.24D6  D12.25D6  S32×Q8  C3⋊S3⋊Dic6  C12.85S32  C335(C2×Q8)  C3⋊S34Dic6
S3×Dic6 is a maximal quotient of
Dic35Dic6  C62.9C23  C62.10C23  Dic36Dic6  Dic3.Dic6  C62.16C23  D6⋊Dic6  D66Dic6  D67Dic6  Dic3⋊Dic6  C62.37C23  D61Dic6  D62Dic6  D63Dic6  D64Dic6  C123Dic6  C3⋊S3⋊Dic6  C335(C2×Q8)  C3⋊S34Dic6

Matrix representation of S3×Dic6 in GL6(𝔽13)

100000
010000
0012100
0012000
000010
000001
,
1200000
0120000
0001200
0012000
000010
000001
,
500000
080000
0012000
0001200
0000012
0000112
,
010000
1200000
001000
000100
000001
000010

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

S3×Dic6 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_6
% in TeX

G:=Group("S3xDic6");
// GroupNames label

G:=SmallGroup(144,137);
// by ID

G=gap.SmallGroup(144,137);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,116,50,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of S3×Dic6 in TeX

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