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## G = Dic6⋊S3order 144 = 24·32

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

Aliases: Dic62S3, D12.2S3, C12.11D6, C323SD16, C4.9S32, (C3×C6).8D4, C32(D4.S3), C324C82C2, (C3×Dic6)⋊1C2, (C3×D12).1C2, C6.8(C3⋊D4), C32(Q82S3), (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 >

Character table of Dic6⋊S3

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 8A 8B 12A 12B 12C 12D 12E 12F size 1 1 12 2 2 4 2 12 2 2 4 12 12 18 18 4 4 4 4 12 12 ρ1 1 1 1 1 1 1 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 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 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 1 1 1 1 1 1 linear of order 2 ρ5 2 2 0 -1 2 -1 2 -2 2 -1 -1 0 0 0 0 -1 2 -1 -1 1 1 orthogonal lifted from D6 ρ6 2 2 0 -1 2 -1 2 2 2 -1 -1 0 0 0 0 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 2 2 -1 -1 2 0 -1 2 -1 -1 -1 0 0 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ8 2 2 -2 2 -1 -1 2 0 -1 2 -1 1 1 0 0 2 -1 -1 -1 0 0 orthogonal lifted from D6 ρ9 2 2 0 2 2 2 -2 0 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 orthogonal lifted from D4 ρ10 2 2 0 -1 2 -1 -2 0 2 -1 -1 0 0 0 0 1 -2 1 1 √-3 -√-3 complex lifted from C3⋊D4 ρ11 2 2 0 2 -1 -1 -2 0 -1 2 -1 √-3 -√-3 0 0 -2 1 1 1 0 0 complex lifted from C3⋊D4 ρ12 2 2 0 -1 2 -1 -2 0 2 -1 -1 0 0 0 0 1 -2 1 1 -√-3 √-3 complex lifted from C3⋊D4 ρ13 2 2 0 2 -1 -1 -2 0 -1 2 -1 -√-3 √-3 0 0 -2 1 1 1 0 0 complex lifted from C3⋊D4 ρ14 2 -2 0 2 2 2 0 0 -2 -2 -2 0 0 √-2 -√-2 0 0 0 0 0 0 complex lifted from SD16 ρ15 2 -2 0 2 2 2 0 0 -2 -2 -2 0 0 -√-2 √-2 0 0 0 0 0 0 complex lifted from SD16 ρ16 4 -4 0 -2 4 -2 0 0 -4 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ17 4 4 0 -2 -2 1 4 0 -2 -2 1 0 0 0 0 -2 -2 1 1 0 0 orthogonal lifted from S32 ρ18 4 4 0 -2 -2 1 -4 0 -2 -2 1 0 0 0 0 2 2 -1 -1 0 0 symplectic lifted from D6⋊S3, Schur index 2 ρ19 4 -4 0 4 -2 -2 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ20 4 -4 0 -2 -2 1 0 0 2 2 -1 0 0 0 0 0 0 3i -3i 0 0 complex faithful ρ21 4 -4 0 -2 -2 1 0 0 2 2 -1 0 0 0 0 0 0 -3i 3i 0 0 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 16 7 22)(2 15 8 21)(3 14 9 20)(4 13 10 19)(5 24 11 18)(6 23 12 17)(25 44 31 38)(26 43 32 37)(27 42 33 48)(28 41 34 47)(29 40 35 46)(30 39 36 45)
(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 32)(3 27)(4 34)(5 29)(6 36)(7 31)(8 26)(9 33)(10 28)(11 35)(12 30)(13 44)(14 39)(15 46)(16 41)(17 48)(18 43)(19 38)(20 45)(21 40)(22 47)(23 42)(24 37)

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

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

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

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

 0 3 0 4 0 4 2 2 1 0 0 0 0 0 1 1
,
 3 4 2 3 1 4 2 1 2 0 1 3 4 4 3 2
,
 2 2 4 1 0 4 3 1 3 2 2 0 1 3 4 0
,
 3 3 1 4 0 1 2 4 3 2 2 0 1 4 1 4
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

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