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## G = C3×Dic18order 216 = 23·33

### Direct product of C3 and Dic18

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C3×Dic18
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C3×Dic9 — C3×Dic18
 Lower central C9 — C18 — C3×Dic18
 Upper central C1 — C6 — C12

Generators and relations for C3×Dic18
G = < a,b,c | a3=b36=1, c2=b18, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×Dic18
On 72 points
Generators in S72
(1 13 25)(2 14 26)(3 15 27)(4 16 28)(5 17 29)(6 18 30)(7 19 31)(8 20 32)(9 21 33)(10 22 34)(11 23 35)(12 24 36)(37 61 49)(38 62 50)(39 63 51)(40 64 52)(41 65 53)(42 66 54)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)
(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 55 19 37)(2 54 20 72)(3 53 21 71)(4 52 22 70)(5 51 23 69)(6 50 24 68)(7 49 25 67)(8 48 26 66)(9 47 27 65)(10 46 28 64)(11 45 29 63)(12 44 30 62)(13 43 31 61)(14 42 32 60)(15 41 33 59)(16 40 34 58)(17 39 35 57)(18 38 36 56)

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

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

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

C3×Dic18 is a maximal subgroup of
C6.D36  C3⋊Dic36  D12.D9  C12.D18  Dic18⋊S3  D12⋊D9  Dic9.D6  C3×Q8×D9

63 conjugacy classes

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

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + + - + - image C1 C2 C2 C3 C6 C6 S3 Q8 D6 D9 C3×S3 C3×Q8 Dic6 D18 S3×C6 C3×D9 Dic18 C3×Dic6 C6×D9 C3×Dic18 kernel C3×Dic18 C3×Dic9 C3×C36 Dic18 Dic9 C36 C3×C12 C3×C9 C3×C6 C12 C12 C9 C32 C6 C6 C4 C3 C3 C2 C1 # reps 1 2 1 2 4 2 1 1 1 3 2 2 2 3 2 6 6 4 6 12

Matrix representation of C3×Dic18 in GL2(𝔽37) generated by

 10 0 0 10
,
 32 0 0 22
,
 0 1 36 0
G:=sub<GL(2,GF(37))| [10,0,0,10],[32,0,0,22],[0,36,1,0] >;

C3×Dic18 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{18}
% in TeX

G:=Group("C3xDic18");
// GroupNames label

G:=SmallGroup(216,43);
// by ID

G=gap.SmallGroup(216,43);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,3604,208,5189]);
// Polycyclic

G:=Group<a,b,c|a^3=b^36=1,c^2=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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