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## G = C6×Dic9order 216 = 23·33

### Direct product of C6 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C6×Dic9
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C3×Dic9 — C6×Dic9
 Lower central C9 — C6×Dic9
 Upper central C1 — C2×C6

Generators and relations for C6×Dic9
G = < a,b,c | a6=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 128 in 58 conjugacy classes, 36 normal (20 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C18, C18, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C3×C9, Dic9, C2×C18, C2×C18, C3×Dic3, C62, C3×C18, C3×C18, C2×Dic9, C6×Dic3, C3×Dic9, C6×C18, C6×Dic9
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, D9, C3×S3, C2×Dic3, C2×C12, Dic9, D18, C3×Dic3, S3×C6, C3×D9, C2×Dic9, C6×Dic3, C3×Dic9, C6×D9, C6×Dic9

Smallest permutation representation of C6×Dic9
On 72 points
Generators in S72
(1 33 13 27 7 21)(2 34 14 28 8 22)(3 35 15 29 9 23)(4 36 16 30 10 24)(5 19 17 31 11 25)(6 20 18 32 12 26)(37 57 43 63 49 69)(38 58 44 64 50 70)(39 59 45 65 51 71)(40 60 46 66 52 72)(41 61 47 67 53 55)(42 62 48 68 54 56)
(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 50 10 41)(2 49 11 40)(3 48 12 39)(4 47 13 38)(5 46 14 37)(6 45 15 54)(7 44 16 53)(8 43 17 52)(9 42 18 51)(19 66 28 57)(20 65 29 56)(21 64 30 55)(22 63 31 72)(23 62 32 71)(24 61 33 70)(25 60 34 69)(26 59 35 68)(27 58 36 67)

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

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

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

C6×Dic9 is a maximal subgroup of   Dic9⋊Dic3  C18.Dic6  Dic3⋊Dic9  C6.18D36  D6⋊Dic9  Dic3.D18  D9×C2×C12

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6O 9A ··· 9I 12A ··· 12H 18A ··· 18AA order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 size 1 1 1 1 1 1 2 2 2 9 9 9 9 1 ··· 1 2 ··· 2 2 ··· 2 9 ··· 9 2 ··· 2

72 irreducible representations

 dim 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 C4 C6 C6 C12 S3 Dic3 D6 D9 C3×S3 Dic9 D18 C3×Dic3 S3×C6 C3×D9 C3×Dic9 C6×D9 kernel C6×Dic9 C3×Dic9 C6×C18 C2×Dic9 C3×C18 Dic9 C2×C18 C18 C62 C3×C6 C3×C6 C2×C6 C2×C6 C6 C6 C6 C6 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 2 1 3 2 6 3 4 2 6 12 6

Matrix representation of C6×Dic9 in GL3(𝔽37) generated by

 27 0 0 0 27 0 0 0 27
,
 1 0 0 0 25 0 0 0 3
,
 36 0 0 0 0 1 0 36 0
G:=sub<GL(3,GF(37))| [27,0,0,0,27,0,0,0,27],[1,0,0,0,25,0,0,0,3],[36,0,0,0,0,36,0,1,0] >;

C6×Dic9 in GAP, Magma, Sage, TeX

C_6\times {\rm Dic}_9
% in TeX

G:=Group("C6xDic9");
// GroupNames label

G:=SmallGroup(216,55);
// by ID

G=gap.SmallGroup(216,55);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,3604,208,5189]);
// Polycyclic

G:=Group<a,b,c|a^6=b^18=1,c^2=b^9,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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