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## G = D6.Dic3order 144 = 24·32

### The non-split extension by D6 of Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D6.Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D6.Dic3
 Lower central C32 — C3×C6 — D6.Dic3
 Upper central C1 — C4

Generators and relations for D6.Dic3
G = < a,b,c,d | a6=b2=1, c6=a3, d2=a3c3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c5 >

Character table of D6.Dic3

 class 1 2A 2B 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 24A 24B 24C 24D size 1 1 6 2 2 4 1 1 6 2 2 4 6 6 6 6 18 18 2 2 2 2 4 4 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 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 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 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 1 1 1 -1 -1 1 1 1 1 linear of order 2 ρ5 1 1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 -i i i -i -1 -1 -1 -1 -1 -1 1 1 i -i -i i linear of order 4 ρ6 1 1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 i -i -i i -1 -1 -1 -1 -1 -1 1 1 -i i i -i linear of order 4 ρ7 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 i -i i -i -1 -1 -1 -1 -1 -1 -1 -1 -i i i -i linear of order 4 ρ8 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 -i i -i i -1 -1 -1 -1 -1 -1 -1 -1 i -i -i i linear of order 4 ρ9 2 2 -2 2 -1 -1 2 2 -2 2 -1 -1 1 1 0 0 0 0 -1 2 2 -1 -1 -1 1 1 0 0 0 0 orthogonal lifted from D6 ρ10 2 2 0 -1 2 -1 2 2 0 -1 2 -1 0 0 2 2 0 0 2 -1 -1 2 -1 -1 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 0 -1 2 -1 2 2 0 -1 2 -1 0 0 -2 -2 0 0 2 -1 -1 2 -1 -1 0 0 1 1 1 1 orthogonal lifted from D6 ρ12 2 2 2 2 -1 -1 2 2 2 2 -1 -1 -1 -1 0 0 0 0 -1 2 2 -1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ13 2 2 -2 2 -1 -1 -2 -2 2 2 -1 -1 1 1 0 0 0 0 1 -2 -2 1 1 1 -1 -1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ14 2 2 2 2 -1 -1 -2 -2 -2 2 -1 -1 -1 -1 0 0 0 0 1 -2 -2 1 1 1 1 1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ15 2 2 0 -1 2 -1 -2 -2 0 -1 2 -1 0 0 -2i 2i 0 0 -2 1 1 -2 1 1 0 0 -i i i -i complex lifted from C4×S3 ρ16 2 2 0 -1 2 -1 -2 -2 0 -1 2 -1 0 0 2i -2i 0 0 -2 1 1 -2 1 1 0 0 i -i -i i complex lifted from C4×S3 ρ17 2 -2 0 2 2 2 2i -2i 0 -2 -2 -2 0 0 0 0 0 0 2i 2i -2i -2i 2i -2i 0 0 0 0 0 0 complex lifted from M4(2) ρ18 2 -2 0 2 2 2 -2i 2i 0 -2 -2 -2 0 0 0 0 0 0 -2i -2i 2i 2i -2i 2i 0 0 0 0 0 0 complex lifted from M4(2) ρ19 2 -2 0 2 -1 -1 2i -2i 0 -2 1 1 √-3 -√-3 0 0 0 0 -i 2i -2i i -i i -√3 √3 0 0 0 0 complex lifted from C4.Dic3 ρ20 2 -2 0 2 -1 -1 -2i 2i 0 -2 1 1 -√-3 √-3 0 0 0 0 i -2i 2i -i i -i -√3 √3 0 0 0 0 complex lifted from C4.Dic3 ρ21 2 -2 0 2 -1 -1 -2i 2i 0 -2 1 1 √-3 -√-3 0 0 0 0 i -2i 2i -i i -i √3 -√3 0 0 0 0 complex lifted from C4.Dic3 ρ22 2 -2 0 2 -1 -1 2i -2i 0 -2 1 1 -√-3 √-3 0 0 0 0 -i 2i -2i i -i i √3 -√3 0 0 0 0 complex lifted from C4.Dic3 ρ23 2 -2 0 -1 2 -1 2i -2i 0 1 -2 1 0 0 0 0 0 0 2i -i i -2i -i i 0 0 2ζ8ζ3+ζ8 2ζ83ζ3+ζ83 2ζ87ζ3+ζ87 2ζ85ζ3+ζ85 complex lifted from C8⋊S3 ρ24 2 -2 0 -1 2 -1 -2i 2i 0 1 -2 1 0 0 0 0 0 0 -2i i -i 2i i -i 0 0 2ζ83ζ3+ζ83 2ζ8ζ3+ζ8 2ζ85ζ3+ζ85 2ζ87ζ3+ζ87 complex lifted from C8⋊S3 ρ25 2 -2 0 -1 2 -1 2i -2i 0 1 -2 1 0 0 0 0 0 0 2i -i i -2i -i i 0 0 2ζ85ζ3+ζ85 2ζ87ζ3+ζ87 2ζ83ζ3+ζ83 2ζ8ζ3+ζ8 complex lifted from C8⋊S3 ρ26 2 -2 0 -1 2 -1 -2i 2i 0 1 -2 1 0 0 0 0 0 0 -2i i -i 2i i -i 0 0 2ζ87ζ3+ζ87 2ζ85ζ3+ζ85 2ζ8ζ3+ζ8 2ζ83ζ3+ζ83 complex lifted from C8⋊S3 ρ27 4 4 0 -2 -2 1 4 4 0 -2 -2 1 0 0 0 0 0 0 -2 -2 -2 -2 1 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ28 4 4 0 -2 -2 1 -4 -4 0 -2 -2 1 0 0 0 0 0 0 2 2 2 2 -1 -1 0 0 0 0 0 0 symplectic lifted from S3×Dic3, Schur index 2 ρ29 4 -4 0 -2 -2 1 4i -4i 0 2 2 -1 0 0 0 0 0 0 -2i -2i 2i 2i i -i 0 0 0 0 0 0 complex faithful ρ30 4 -4 0 -2 -2 1 -4i 4i 0 2 2 -1 0 0 0 0 0 0 2i 2i -2i -2i -i i 0 0 0 0 0 0 complex faithful

Smallest permutation representation of D6.Dic3
On 48 points
Generators in S48
(1 11 9 7 5 3)(2 12 10 8 6 4)(13 15 17 19 21 23)(14 16 18 20 22 24)(25 27 29 31 33 35)(26 28 30 32 34 36)(37 47 45 43 41 39)(38 48 46 44 42 40)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 37)(23 38)(24 39)
(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 19 10 16 7 13 4 22)(2 24 11 21 8 18 5 15)(3 17 12 14 9 23 6 20)(25 41 34 38 31 47 28 44)(26 46 35 43 32 40 29 37)(27 39 36 48 33 45 30 42)

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

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

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

Matrix representation of D6.Dic3 in GL4(𝔽5) generated by

 4 0 2 0 0 4 0 2 1 0 2 0 0 1 0 2
,
 0 1 0 4 4 0 1 0 0 2 0 4 3 0 1 0
,
 2 0 1 0 0 1 0 4 3 0 1 0 0 2 0 2
,
 0 1 0 1 0 0 4 0 0 3 0 0 2 0 1 0
G:=sub<GL(4,GF(5))| [4,0,1,0,0,4,0,1,2,0,2,0,0,2,0,2],[0,4,0,3,1,0,2,0,0,1,0,1,4,0,4,0],[2,0,3,0,0,1,0,2,1,0,1,0,0,4,0,2],[0,0,0,2,1,0,3,0,0,4,0,1,1,0,0,0] >;

D6.Dic3 in GAP, Magma, Sage, TeX

D_6.{\rm Dic}_3
% in TeX

G:=Group("D6.Dic3");
// GroupNames label

G:=SmallGroup(144,54);
// by ID

G=gap.SmallGroup(144,54);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,31,50,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=1,c^6=a^3,d^2=a^3*c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^5>;
// generators/relations

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