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## G = S3×Dic9order 216 = 23·33

### Direct product of S3 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — S3×Dic9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — S3×C18 — S3×Dic9
 Lower central C3×C9 — S3×Dic9
 Upper central C1 — C2

Generators and relations for S3×Dic9
G = < a,b,c,d | a3=b2=c18=1, d2=c9, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of S3×Dic9
On 72 points
Generators in S72
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)
(1 52)(2 53)(3 54)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 71)(20 72)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)
(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 29 10 20)(2 28 11 19)(3 27 12 36)(4 26 13 35)(5 25 14 34)(6 24 15 33)(7 23 16 32)(8 22 17 31)(9 21 18 30)(37 60 46 69)(38 59 47 68)(39 58 48 67)(40 57 49 66)(41 56 50 65)(42 55 51 64)(43 72 52 63)(44 71 53 62)(45 70 54 61)

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

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

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

S3×Dic9 is a maximal subgroup of   D125D9  D12⋊D9  C4×S3×D9  Dic3.D18  D18.4D6
S3×Dic9 is a maximal quotient of   D6.Dic9  Dic3⋊Dic9  D6⋊Dic9

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 18G ··· 18L 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 18 ··· 18 size 1 1 3 3 2 2 4 9 9 27 27 2 2 4 6 6 2 2 2 4 4 4 18 18 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 D6 Dic3 D6 D9 C4×S3 Dic9 D18 S32 S3×Dic3 S3×D9 S3×Dic9 kernel S3×Dic9 C3×Dic9 C9⋊Dic3 S3×C18 S3×C9 Dic9 S3×C6 C18 C3×S3 C3×C6 D6 C9 S3 C6 C6 C3 C2 C1 # reps 1 1 1 1 4 1 1 1 2 1 3 2 6 3 1 1 3 3

Matrix representation of S3×Dic9 in GL4(𝔽37) generated by

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

S3×Dic9 in GAP, Magma, Sage, TeX

S_3\times {\rm Dic}_9
% in TeX

G:=Group("S3xDic9");
// GroupNames label

G:=SmallGroup(216,30);
// by ID

G=gap.SmallGroup(216,30);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,1065,453,1444,2603]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^18=1,d^2=c^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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