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G = Dic6⋊S3  order 144 = 24·32

2nd semidirect product of Dic6 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: Dic6⋊2S3, D12.2S3, C12.11D6, C32⋊3SD16, C4.9S32, (C3×C6).8D4, C3⋊2(D4.S3), C32⋊4C8⋊2C2, (C3×Dic6)⋊1C2, (C3×D12).1C2, C6.8(C3⋊D4), C3⋊2(Q8⋊2S3), (C3×C12).3C22, C2.4(D6⋊S3), SmallGroup(144,58)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — Dic6⋊S3
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — Dic6⋊S3
Lower central
C32 — C3×C6 — C3×C12 — Dic6⋊S3
Upper central
C1 — C2 — C4

Generators and relations for Dic6⋊S3
 G = < a,b,c,d | a12=c3=d2=1, b2=a6, bab-1=a-1, ac=ca, dad=a7, bc=cb, dbd=a3b, dcd=c-1 >

C1
C2
12C2
C3
C3
2C3
C4
6C22
6C4
C6
C6
2C6
4S3
12C6
C32
3Q8
3D4
9C8
C12
C12
2C12
2D6
2Dic3
6C12
6C2×C6
C3×C6
4C3×S3
9SD16
Dic6
D12
3C3×Q8
3C3×D4
3C3⋊C8
3C3⋊C8
6C3⋊C8
C3×C12
2S3×C6
2C3×Dic3
3Q8⋊2S3
3D4.S3
C3×D12
C32⋊4C8
C3×Dic6
Dic6⋊S3

Character table of Dic6⋊S3

 class 12A2B3A3B3C4A4B6A6B6C6D6E8A8B12A12B12C12D12E12F
 size 11122242122241212181844441212
ρ1111111111111111111111    trivial
ρ21111111-111111-1-11111-1-1    linear of order 2
ρ311-11111-1111-1-1111111-1-1    linear of order 2
ρ411-111111111-1-1-1-1111111    linear of order 2
ρ5220-12-12-22-1-10000-12-1-111    orthogonal lifted from D6
ρ6220-12-1222-1-10000-12-1-1-1-1    orthogonal lifted from S3
ρ72222-1-120-12-1-1-1002-1-1-100    orthogonal lifted from S3
ρ822-22-1-120-12-111002-1-1-100    orthogonal lifted from D6
ρ9220222-202220000-2-2-2-200    orthogonal lifted from D4
ρ10220-12-1-202-1-100001-211√-3-√-3    complex lifted from C3⋊D4
ρ112202-1-1-20-12-1√-3-√-300-211100    complex lifted from C3⋊D4
ρ12220-12-1-202-1-100001-211-√-3√-3    complex lifted from C3⋊D4
ρ132202-1-1-20-12-1-√-3√-300-211100    complex lifted from C3⋊D4
ρ142-2022200-2-2-200√-2-√-2000000    complex lifted from SD16
ρ152-2022200-2-2-200-√-2√-2000000    complex lifted from SD16
ρ164-40-24-200-4220000000000    orthogonal lifted from Q8⋊2S3
ρ17440-2-2140-2-210000-2-21100    orthogonal lifted from S32
ρ18440-2-21-40-2-21000022-1-100    symplectic lifted from D6⋊S3, Schur index 2
ρ194-404-2-2002-420000000000    symplectic lifted from D4.S3, Schur index 2
ρ204-40-2-210022-10000003i-3i00    complex faithful
ρ214-40-2-210022-1000000-3i3i00    complex faithful

Smallest permutation representation of Dic6⋊S3
►On 48 points
Generators in S48
(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 38 31 44)(26 37 32 43)(27 48 33 42)(28 47 34 41)(29 46 35 40)(30 45 36 39)

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

Dic6⋊S3 is a maximal subgroup of
 C24⋊9D6  C24⋊6D6  D24⋊5S3  D12.4D6  D12.30D6  D12⋊20D6  D12.32D6  Dic6⋊3D6  S3×D4.S3  D12.22D6  D12.7D6  S3×Q8⋊2S3  D12.11D6  D12.24D6  D12.13D6  D12.D9  Dic6⋊D9  He3⋊4SD16  He3⋊5SD16  C33⋊12SD16  C33⋊13SD16  C33⋊18SD16
Dic6⋊S3 is a maximal quotient of
 D12⋊3Dic3  Dic6⋊Dic3  C12.6Dic6  D12.D9  Dic6⋊D9  He3⋊3SD16  C33⋊12SD16  C33⋊13SD16  C33⋊18SD16

Matrix representation of Dic6⋊S3 ►in GL4(𝔽5) generated by

0304
0422
1000
0011
,
3423
1421
2013
4432
,
2241
0431
3220
1340
,
3314
0124
3220
1414
G:=sub<GL(4,GF(5))| [0,0,1,0,3,4,0,0,0,2,0,1,4,2,0,1],[3,1,2,4,4,4,0,4,2,2,1,3,3,1,3,2],[2,0,3,1,2,4,2,3,4,3,2,4,1,1,0,0],[3,0,3,1,3,1,2,4,1,2,2,1,4,4,0,4] >;
 

Dic6⋊S3 in GAP, Magma, Sage, TeX 

{\rm Dic}_6\rtimes S_3
 
% in TeX
 
G:=Group("Dic6:S3");
 
// GroupNames label
 
G:=SmallGroup(144,58);
 
// by ID
 
G=gap.SmallGroup(144,58);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,55,218,116,50,490,3461]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=c^3=d^2=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^7,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of Dic6⋊S3 in TeX
Character table of Dic6⋊S3 in TeX

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