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## G = Dic3×D9order 216 = 23·33

### Direct product of Dic3 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — Dic3×D9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — Dic3×D9
 Lower central C3×C9 — Dic3×D9
 Upper central C1 — C2

Generators and relations for Dic3×D9
G = < a,b,c,d | a6=c9=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of Dic3×D9
On 72 points
Generators in S72
(1 17 7 14 4 11)(2 18 8 15 5 12)(3 10 9 16 6 13)(19 34 22 28 25 31)(20 35 23 29 26 32)(21 36 24 30 27 33)(37 52 40 46 43 49)(38 53 41 47 44 50)(39 54 42 48 45 51)(55 67 61 64 58 70)(56 68 62 65 59 71)(57 69 63 66 60 72)
(1 50 14 41)(2 51 15 42)(3 52 16 43)(4 53 17 44)(5 54 18 45)(6 46 10 37)(7 47 11 38)(8 48 12 39)(9 49 13 40)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)
(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 49 50 51 52 53 54)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)
(1 30)(2 29)(3 28)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 25)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(17 27)(18 26)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 64)(44 72)(45 71)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 63)(54 62)

G:=sub<Sym(72)| (1,17,7,14,4,11)(2,18,8,15,5,12)(3,10,9,16,6,13)(19,34,22,28,25,31)(20,35,23,29,26,32)(21,36,24,30,27,33)(37,52,40,46,43,49)(38,53,41,47,44,50)(39,54,42,48,45,51)(55,67,61,64,58,70)(56,68,62,65,59,71)(57,69,63,66,60,72), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (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)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,30)(2,29)(3,28)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,27)(18,26)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,72)(45,71)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,63)(54,62)>;

G:=Group( (1,17,7,14,4,11)(2,18,8,15,5,12)(3,10,9,16,6,13)(19,34,22,28,25,31)(20,35,23,29,26,32)(21,36,24,30,27,33)(37,52,40,46,43,49)(38,53,41,47,44,50)(39,54,42,48,45,51)(55,67,61,64,58,70)(56,68,62,65,59,71)(57,69,63,66,60,72), (1,50,14,41)(2,51,15,42)(3,52,16,43)(4,53,17,44)(5,54,18,45)(6,46,10,37)(7,47,11,38)(8,48,12,39)(9,49,13,40)(19,64,28,55)(20,65,29,56)(21,66,30,57)(22,67,31,58)(23,68,32,59)(24,69,33,60)(25,70,34,61)(26,71,35,62)(27,72,36,63), (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)(46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,30)(2,29)(3,28)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,27)(18,26)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,72)(45,71)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,63)(54,62) );

G=PermutationGroup([[(1,17,7,14,4,11),(2,18,8,15,5,12),(3,10,9,16,6,13),(19,34,22,28,25,31),(20,35,23,29,26,32),(21,36,24,30,27,33),(37,52,40,46,43,49),(38,53,41,47,44,50),(39,54,42,48,45,51),(55,67,61,64,58,70),(56,68,62,65,59,71),(57,69,63,66,60,72)], [(1,50,14,41),(2,51,15,42),(3,52,16,43),(4,53,17,44),(5,54,18,45),(6,46,10,37),(7,47,11,38),(8,48,12,39),(9,49,13,40),(19,64,28,55),(20,65,29,56),(21,66,30,57),(22,67,31,58),(23,68,32,59),(24,69,33,60),(25,70,34,61),(26,71,35,62),(27,72,36,63)], [(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),(46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(1,30),(2,29),(3,28),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,25),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19),(17,27),(18,26),(37,70),(38,69),(39,68),(40,67),(41,66),(42,65),(43,64),(44,72),(45,71),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,63),(54,62)]])

Dic3×D9 is a maximal subgroup of   D18.D6  D365S3  C4×S3×D9  D18.3D6  D18.4D6
Dic3×D9 is a maximal quotient of   C36.39D6  Dic9⋊Dic3  D18⋊Dic3

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 9A 9B 9C 9D 9E 9F 12A 12B 18A 18B 18C 18D 18E 18F 36A ··· 36F order 1 2 2 2 3 3 3 4 4 4 4 6 6 6 6 6 9 9 9 9 9 9 12 12 18 18 18 18 18 18 36 ··· 36 size 1 1 9 9 2 2 4 3 3 27 27 2 2 4 18 18 2 2 2 4 4 4 6 6 2 2 2 4 4 4 6 ··· 6

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + + + + - + - image C1 C2 C2 C2 C4 S3 S3 Dic3 D6 D6 D9 C4×S3 D18 C4×D9 S32 S3×Dic3 S3×D9 Dic3×D9 kernel Dic3×D9 C9×Dic3 C9⋊Dic3 C6×D9 C3×D9 D18 C3×Dic3 D9 C18 C3×C6 Dic3 C32 C6 C3 C6 C3 C2 C1 # reps 1 1 1 1 4 1 1 2 1 1 3 2 3 6 1 1 3 3

Matrix representation of Dic3×D9 in GL4(𝔽37) generated by

 0 1 0 0 36 1 0 0 0 0 1 0 0 0 0 1
,
 0 6 0 0 6 0 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 1 0 0 0 0 20 31 0 0 6 26
,
 1 0 0 0 0 1 0 0 0 0 20 31 0 0 11 17
G:=sub<GL(4,GF(37))| [0,36,0,0,1,1,0,0,0,0,1,0,0,0,0,1],[0,6,0,0,6,0,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,20,6,0,0,31,26],[1,0,0,0,0,1,0,0,0,0,20,11,0,0,31,17] >;

Dic3×D9 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times D_9
% in TeX

G:=Group("Dic3xD9");
// GroupNames label

G:=SmallGroup(216,27);
// by ID

G=gap.SmallGroup(216,27);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,1065,453,1444,2603]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^9=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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