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## G = Dic3⋊Dic3order 144 = 24·32

### The semidirect product of Dic3 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — Dic3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic3⋊Dic3
 Lower central C32 — C3×C6 — Dic3⋊Dic3
 Upper central C1 — C22

Generators and relations for Dic3⋊Dic3
G = < a,b,c,d | a6=c6=1, b2=a3, d2=c3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 160 in 60 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×Dic3, C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C4⋊Dic3, C6×Dic3, C2×C3⋊Dic3, Dic3⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, S32, Dic3⋊C4, C4⋊Dic3, S3×Dic3, C3⋊D12, C322Q8, Dic3⋊Dic3

Character table of Dic3⋊Dic3

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 2 2 4 6 6 6 6 18 18 2 2 2 2 2 2 4 4 4 6 6 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 i 1 -i i -i -1 -1 1 -1 -1 1 -1 1 -1 -i 1 1 -i -1 -1 i i linear of order 4 ρ6 1 -1 1 -1 1 1 1 -1 -i 1 i -i i -1 -1 1 -1 -1 1 -1 1 -1 i 1 1 i -1 -1 -i -i linear of order 4 ρ7 1 -1 1 -1 1 1 1 1 i -1 -i -i i -1 -1 1 -1 -1 1 -1 1 -1 -i -1 -1 -i 1 1 i i linear of order 4 ρ8 1 -1 1 -1 1 1 1 1 -i -1 i i -i -1 -1 1 -1 -1 1 -1 1 -1 i -1 -1 i 1 1 -i -i linear of order 4 ρ9 2 2 2 2 -1 2 -1 0 2 0 2 0 0 -1 2 -1 -1 2 2 -1 -1 -1 -1 0 0 -1 0 0 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 2 -1 -1 -2 0 -2 0 0 0 2 -1 2 2 -1 -1 -1 -1 -1 0 1 1 0 1 1 0 0 orthogonal lifted from D6 ρ12 2 2 2 2 2 -1 -1 2 0 2 0 0 0 2 -1 2 2 -1 -1 -1 -1 -1 0 -1 -1 0 -1 -1 0 0 orthogonal lifted from S3 ρ13 2 2 -2 -2 2 -1 -1 0 0 0 0 0 0 -2 1 -2 2 -1 1 1 1 -1 0 √3 -√3 0 √3 -√3 0 0 orthogonal lifted from D12 ρ14 2 2 2 2 -1 2 -1 0 -2 0 -2 0 0 -1 2 -1 -1 2 2 -1 -1 -1 1 0 0 1 0 0 1 1 orthogonal lifted from D6 ρ15 2 2 -2 -2 2 -1 -1 0 0 0 0 0 0 -2 1 -2 2 -1 1 1 1 -1 0 -√3 √3 0 -√3 √3 0 0 orthogonal lifted from D12 ρ16 2 -2 2 -2 2 -1 -1 -2 0 2 0 0 0 -2 1 2 -2 1 -1 1 -1 1 0 -1 -1 0 1 1 0 0 symplectic lifted from Dic3, Schur index 2 ρ17 2 -2 2 -2 2 -1 -1 2 0 -2 0 0 0 -2 1 2 -2 1 -1 1 -1 1 0 1 1 0 -1 -1 0 0 symplectic lifted from Dic3, Schur index 2 ρ18 2 -2 -2 2 2 2 2 0 0 0 0 0 0 2 2 -2 -2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ19 2 -2 -2 2 -1 2 -1 0 0 0 0 0 0 -1 2 1 1 -2 -2 -1 1 1 -√3 0 0 √3 0 0 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ20 2 -2 -2 2 2 -1 -1 0 0 0 0 0 0 2 -1 -2 -2 1 1 -1 1 1 0 √3 -√3 0 -√3 √3 0 0 symplectic lifted from Dic6, Schur index 2 ρ21 2 -2 -2 2 -1 2 -1 0 0 0 0 0 0 -1 2 1 1 -2 -2 -1 1 1 √3 0 0 -√3 0 0 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ22 2 -2 -2 2 2 -1 -1 0 0 0 0 0 0 2 -1 -2 -2 1 1 -1 1 1 0 -√3 √3 0 √3 -√3 0 0 symplectic lifted from Dic6, Schur index 2 ρ23 2 -2 2 -2 -1 2 -1 0 -2i 0 2i 0 0 1 -2 -1 1 -2 2 1 -1 1 -i 0 0 -i 0 0 i i complex lifted from C4×S3 ρ24 2 -2 2 -2 -1 2 -1 0 2i 0 -2i 0 0 1 -2 -1 1 -2 2 1 -1 1 i 0 0 i 0 0 -i -i complex lifted from C4×S3 ρ25 2 2 -2 -2 -1 2 -1 0 0 0 0 0 0 1 -2 1 -1 2 -2 1 1 -1 -√-3 0 0 √-3 0 0 -√-3 √-3 complex lifted from C3⋊D4 ρ26 2 2 -2 -2 -1 2 -1 0 0 0 0 0 0 1 -2 1 -1 2 -2 1 1 -1 √-3 0 0 -√-3 0 0 √-3 -√-3 complex lifted from C3⋊D4 ρ27 4 4 -4 -4 -2 -2 1 0 0 0 0 0 0 2 2 2 -2 -2 2 -1 -1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C3⋊D12 ρ28 4 4 4 4 -2 -2 1 0 0 0 0 0 0 -2 -2 -2 -2 -2 -2 1 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from S32 ρ29 4 -4 -4 4 -2 -2 1 0 0 0 0 0 0 -2 -2 2 2 2 2 1 -1 -1 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊2Q8, Schur index 2 ρ30 4 -4 4 -4 -2 -2 1 0 0 0 0 0 0 2 2 -2 2 2 -2 -1 1 -1 0 0 0 0 0 0 0 0 symplectic lifted from S3×Dic3, Schur index 2

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

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

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

Matrix representation of Dic3⋊Dic3 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 12 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

Dic3⋊Dic3 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes {\rm Dic}_3
% in TeX

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

G:=SmallGroup(144,66);
// by ID

G=gap.SmallGroup(144,66);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,121,31,490,3461]);
// Polycyclic

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

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