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## G = C3×Dic9order 108 = 22·33

### Direct product of C3 and Dic9

Aliases: C3×Dic9, C93C12, C6.4D9, C18.3C6, C32.2Dic3, (C3×C9)⋊2C4, C2.(C3×D9), C6.1(C3×S3), (C3×C6).5S3, (C3×C18).2C2, C3.1(C3×Dic3), SmallGroup(108,6)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×Dic9
G = < a,b,c | a3=b18=1, c2=b9, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×Dic9
On 36 points
Generators in S36
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)
(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)
(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)

G:=sub<Sym(36)| (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36), (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), (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)>;

G:=Group( (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36), (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), (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) );

G=PermutationGroup([(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36)], [(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)], [(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)])

C3×Dic9 is a maximal subgroup of
C9⋊Dic6  C18.D6  C9⋊D12  C12×D9  C9⋊C36  C27⋊C12  C322Dic9  He3.3Dic3  He3⋊Dic3  3- 1+2.Dic3  He3.4Dic3  Dic9.2A4
C3×Dic9 is a maximal quotient of
C32⋊Dic9  C27⋊C12

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - + - image C1 C2 C3 C4 C6 C12 S3 Dic3 D9 C3×S3 Dic9 C3×Dic3 C3×D9 C3×Dic9 kernel C3×Dic9 C3×C18 Dic9 C3×C9 C18 C9 C3×C6 C32 C6 C6 C3 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 3 2 3 2 6 6

Matrix representation of C3×Dic9 in GL2(𝔽19) generated by

 7 0 0 7
,
 15 0 0 14
,
 0 18 1 0
G:=sub<GL(2,GF(19))| [7,0,0,7],[15,0,0,14],[0,1,18,0] >;

C3×Dic9 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_9
% in TeX

G:=Group("C3xDic9");
// GroupNames label

G:=SmallGroup(108,6);
// by ID

G=gap.SmallGroup(108,6);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-3,30,1203,138,1804]);
// Polycyclic

G:=Group<a,b,c|a^3=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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