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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 [×2], C3 [×2], C3, C4, C4 [×5], C22, S3 [×2], C6 [×2], C6 [×3], C2×C4 [×3], Q8 [×4], C32, Dic3, Dic3 [×2], Dic3 [×6], C12 [×2], C12 [×4], D6, C2×C6, C2×Q8, C3×S3 [×2], C3×C6, Dic6, Dic6 [×7], C4×S3, C4×S3 [×2], C2×Dic3 [×2], C2×C12, C3×Q8, C3×Dic3, C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12, S3×C6, C2×Dic6, S3×Q8, S3×Dic3 [×2], C322Q8 [×2], C3×Dic6, S3×C12, C324Q8, S3×Dic6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], Q8 [×2], C23, D6 [×6], C2×Q8, Dic6 [×2], C22×S3 [×2], 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 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)
(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 23 7 17)(2 22 8 16)(3 21 9 15)(4 20 10 14)(5 19 11 13)(6 18 12 24)(25 41 31 47)(26 40 32 46)(27 39 33 45)(28 38 34 44)(29 37 35 43)(30 48 36 42)

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

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

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

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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