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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 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)]])

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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