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## G = D6⋊Dic3order 144 = 24·32

### The semidirect product of D6 and 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 — C62 — S3×C2×C6 — D6⋊Dic3
 Lower central C32 — C3×C6 — D6⋊Dic3
 Upper central C1 — C22

Generators and relations for D6⋊Dic3
G = < a,b,c,d | a6=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 224 in 76 conjugacy classes, 30 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C2×C4 [×2], C23, C32, Dic3 [×5], C12, D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C22⋊C4, C3×S3 [×2], C3×C6 [×3], C2×Dic3, C2×Dic3 [×3], C2×C12, C22×S3, C22×C6, C3×Dic3, C3⋊Dic3, S3×C6 [×2], S3×C6 [×2], C62, D6⋊C4, C6.D4, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, D6⋊Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4 [×3], S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, D6⋊Dic3

Character table of D6⋊Dic3

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D size 1 1 1 1 6 6 2 2 4 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 1 i -i i -i -1 -1 1 -1 -1 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 -i i -i i -1 -1 1 -1 -1 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 i -i -i i -1 -1 1 -1 -1 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 -i i i -i -1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 -i -i i i linear of order 4 ρ9 2 2 2 2 0 0 -1 2 -1 2 2 0 0 -1 2 -1 -1 2 2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 -2 -2 2 -1 -1 0 0 0 0 2 -1 2 2 -1 -1 -1 -1 -1 1 1 1 1 0 0 0 0 orthogonal lifted from D6 ρ11 2 2 2 2 2 2 2 -1 -1 0 0 0 0 2 -1 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ12 2 -2 -2 2 0 0 2 2 2 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 ρ13 2 2 -2 -2 0 0 2 2 2 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 ρ14 2 2 2 2 0 0 -1 2 -1 -2 -2 0 0 -1 2 -1 -1 2 2 -1 -1 -1 0 0 0 0 1 1 1 1 orthogonal lifted from D6 ρ15 2 -2 -2 2 0 0 -1 2 -1 0 0 0 0 -1 2 1 1 -2 -2 1 1 -1 0 0 0 0 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ16 2 -2 -2 2 0 0 -1 2 -1 0 0 0 0 -1 2 1 1 -2 -2 1 1 -1 0 0 0 0 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ17 2 -2 2 -2 -2 2 2 -1 -1 0 0 0 0 -2 1 2 -2 1 -1 1 -1 1 -1 1 1 -1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ18 2 -2 2 -2 2 -2 2 -1 -1 0 0 0 0 -2 1 2 -2 1 -1 1 -1 1 1 -1 -1 1 0 0 0 0 symplectic lifted from Dic3, Schur index 2 ρ19 2 -2 -2 2 0 0 2 -1 -1 0 0 0 0 2 -1 -2 -2 1 1 1 1 -1 √-3 √-3 -√-3 -√-3 0 0 0 0 complex lifted from C3⋊D4 ρ20 2 2 -2 -2 0 0 2 -1 -1 0 0 0 0 -2 1 -2 2 -1 1 -1 1 1 -√-3 √-3 -√-3 √-3 0 0 0 0 complex lifted from C3⋊D4 ρ21 2 2 -2 -2 0 0 -1 2 -1 0 0 0 0 1 -2 1 -1 2 -2 -1 1 1 0 0 0 0 √-3 -√-3 -√-3 √-3 complex lifted from C3⋊D4 ρ22 2 2 -2 -2 0 0 2 -1 -1 0 0 0 0 -2 1 -2 2 -1 1 -1 1 1 √-3 -√-3 √-3 -√-3 0 0 0 0 complex lifted from C3⋊D4 ρ23 2 -2 2 -2 0 0 -1 2 -1 -2i 2i 0 0 1 -2 -1 1 -2 2 1 -1 1 0 0 0 0 i i -i -i complex lifted from C4×S3 ρ24 2 -2 2 -2 0 0 -1 2 -1 2i -2i 0 0 1 -2 -1 1 -2 2 1 -1 1 0 0 0 0 -i -i i i complex lifted from C4×S3 ρ25 2 -2 -2 2 0 0 2 -1 -1 0 0 0 0 2 -1 -2 -2 1 1 1 1 -1 -√-3 -√-3 √-3 √-3 0 0 0 0 complex lifted from C3⋊D4 ρ26 2 2 -2 -2 0 0 -1 2 -1 0 0 0 0 1 -2 1 -1 2 -2 -1 1 1 0 0 0 0 -√-3 √-3 √-3 -√-3 complex lifted from C3⋊D4 ρ27 4 -4 -4 4 0 0 -2 -2 1 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 0 0 -2 -2 1 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 0 0 -2 -2 1 0 0 0 0 2 2 2 -2 -2 2 1 -1 -1 0 0 0 0 0 0 0 0 symplectic lifted from D6⋊S3, Schur index 2 ρ30 4 -4 4 -4 0 0 -2 -2 1 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 D6⋊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)(2 33)(3 32)(4 31)(5 36)(6 35)(7 28)(8 27)(9 26)(10 25)(11 30)(12 29)(13 46)(14 45)(15 44)(16 43)(17 48)(18 47)(19 40)(20 39)(21 38)(22 37)(23 42)(24 41)
(1 11 3 7 5 9)(2 12 4 8 6 10)(13 21 17 19 15 23)(14 22 18 20 16 24)(25 33 29 31 27 35)(26 34 30 32 28 36)(37 47 39 43 41 45)(38 48 40 44 42 46)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(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)(2,33)(3,32)(4,31)(5,36)(6,35)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,46)(14,45)(15,44)(16,43)(17,48)(18,47)(19,40)(20,39)(21,38)(22,37)(23,42)(24,41), (1,11,3,7,5,9)(2,12,4,8,6,10)(13,21,17,19,15,23)(14,22,18,20,16,24)(25,33,29,31,27,35)(26,34,30,32,28,36)(37,47,39,43,41,45)(38,48,40,44,42,46), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(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)(2,33)(3,32)(4,31)(5,36)(6,35)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,46)(14,45)(15,44)(16,43)(17,48)(18,47)(19,40)(20,39)(21,38)(22,37)(23,42)(24,41), (1,11,3,7,5,9)(2,12,4,8,6,10)(13,21,17,19,15,23)(14,22,18,20,16,24)(25,33,29,31,27,35)(26,34,30,32,28,36)(37,47,39,43,41,45)(38,48,40,44,42,46), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(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),(2,33),(3,32),(4,31),(5,36),(6,35),(7,28),(8,27),(9,26),(10,25),(11,30),(12,29),(13,46),(14,45),(15,44),(16,43),(17,48),(18,47),(19,40),(20,39),(21,38),(22,37),(23,42),(24,41)], [(1,11,3,7,5,9),(2,12,4,8,6,10),(13,21,17,19,15,23),(14,22,18,20,16,24),(25,33,29,31,27,35),(26,34,30,32,28,36),(37,47,39,43,41,45),(38,48,40,44,42,46)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18),(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 D6⋊Dic3 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 12 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D6⋊Dic3 in GAP, Magma, Sage, TeX

D_6\rtimes {\rm Dic}_3
% in TeX

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

G:=SmallGroup(144,64);
// by ID

G=gap.SmallGroup(144,64);
# by ID

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

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

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