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## G = C9×Dic6order 216 = 23·33

### Direct product of C9 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C9×Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — C9×Dic3 — C9×Dic6
 Lower central C3 — C6 — C9×Dic6
 Upper central C1 — C18 — C36

Generators and relations for C9×Dic6
G = < a,b,c | a9=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C9×Dic6
On 72 points
Generators in S72
(1 44 25 9 40 33 5 48 29)(2 45 26 10 41 34 6 37 30)(3 46 27 11 42 35 7 38 31)(4 47 28 12 43 36 8 39 32)(13 59 61 17 51 65 21 55 69)(14 60 62 18 52 66 22 56 70)(15 49 63 19 53 67 23 57 71)(16 50 64 20 54 68 24 58 72)
(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 67 7 61)(2 66 8 72)(3 65 9 71)(4 64 10 70)(5 63 11 69)(6 62 12 68)(13 48 19 42)(14 47 20 41)(15 46 21 40)(16 45 22 39)(17 44 23 38)(18 43 24 37)(25 57 31 51)(26 56 32 50)(27 55 33 49)(28 54 34 60)(29 53 35 59)(30 52 36 58)

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

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

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

C9×Dic6 is a maximal subgroup of
Dic6⋊D9  C18.D12  C12.D18  C9⋊Dic12  D18.D6  Dic65D9  Dic18⋊S3  S3×Q8×C9

81 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 9A ··· 9F 9G ··· 9L 12A ··· 12H 12I 12J 12K 12L 18A ··· 18F 18G ··· 18L 36A ··· 36R 36S ··· 36AD order 1 2 3 3 3 3 3 4 4 4 6 6 6 6 6 9 ··· 9 9 ··· 9 12 ··· 12 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 1 1 2 2 2 2 6 6 1 1 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C6 C6 C9 C18 C18 S3 Q8 D6 C3×S3 Dic6 C3×Q8 S3×C6 S3×C9 Q8×C9 C3×Dic6 S3×C18 C9×Dic6 kernel C9×Dic6 C9×Dic3 C3×C36 C3×Dic6 C3×Dic3 C3×C12 Dic6 Dic3 C12 C36 C3×C9 C18 C12 C9 C32 C6 C4 C3 C3 C2 C1 # reps 1 2 1 2 4 2 6 12 6 1 1 1 2 2 2 2 6 6 4 6 12

Matrix representation of C9×Dic6 in GL2(𝔽37) generated by

 16 0 0 16
,
 14 0 19 8
,
 27 9 34 10
G:=sub<GL(2,GF(37))| [16,0,0,16],[14,19,0,8],[27,34,9,10] >;

C9×Dic6 in GAP, Magma, Sage, TeX

C_9\times {\rm Dic}_6
% in TeX

G:=Group("C9xDic6");
// GroupNames label

G:=SmallGroup(216,44);
// by ID

G=gap.SmallGroup(216,44);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,122,5189]);
// Polycyclic

G:=Group<a,b,c|a^9=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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