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G = C9×Dic3order 108 = 22·33

Direct product of C9 and Dic3

Aliases: C9×Dic3, C3⋊C36, C6.C18, C18.4S3, C32.2C12, (C3×C9)⋊1C4, C2.(S3×C9), C6.8(C3×S3), (C3×C6).5C6, (C3×C18).1C2, C18(C3×Dic3), (C3×Dic3).C3, C3.4(C3×Dic3), SmallGroup(108,7)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C9×Dic3
G = < a,b,c | a9=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C9×Dic3
On 36 points
Generators in S36
(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 24 4 27 7 21)(2 25 5 19 8 22)(3 26 6 20 9 23)(10 34 16 31 13 28)(11 35 17 32 14 29)(12 36 18 33 15 30)
(1 10 27 31)(2 11 19 32)(3 12 20 33)(4 13 21 34)(5 14 22 35)(6 15 23 36)(7 16 24 28)(8 17 25 29)(9 18 26 30)

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

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

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

C9×Dic3 is a maximal subgroup of
C9⋊Dic6  C18.D6  C3⋊D36  S3×C36  C32⋊C36  C9⋊C36  He3.C12  He3.2C12  He3.5C12  Q8⋊C93S3
C9×Dic3 is a maximal quotient of
C32⋊C36  C9⋊C36

54 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 18A ··· 18F 18G ··· 18L 36A ··· 36L order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 9 ··· 9 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 3 3 1 1 2 2 2 1 ··· 1 2 ··· 2 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3

54 irreducible representations

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

Matrix representation of C9×Dic3 in GL2(𝔽19) generated by

 9 0 0 9
,
 12 0 0 8
,
 0 18 1 0
G:=sub<GL(2,GF(19))| [9,0,0,9],[12,0,0,8],[0,1,18,0] >;

C9×Dic3 in GAP, Magma, Sage, TeX

C_9\times {\rm Dic}_3
% in TeX

G:=Group("C9xDic3");
// GroupNames label

G:=SmallGroup(108,7);
// by ID

G=gap.SmallGroup(108,7);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-3,30,66,1804]);
// Polycyclic

G:=Group<a,b,c|a^9=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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