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G = C3×Dic9order 108 = 22·33

Direct product of C3 and Dic9

direct product, metacyclic, supersoluble, monomial, A-group

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

C1C9 — C3×Dic9
C1C3C9C18C3×C18 — C3×Dic9
C9 — C3×Dic9
C1C6

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

2C3
9C4
2C6
2C9
3Dic3
9C12
2C18
3C3×Dic3

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 3A3B3C3D3E4A4B6A6B6C6D6E9A···9I12A12B12C12D18A···18I
order123333344666669···91212121218···18
size111122299112222···299992···2

36 irreducible representations

dim11111122222222
type+++-+-
imageC1C2C3C4C6C12S3Dic3D9C3×S3Dic9C3×Dic3C3×D9C3×Dic9
kernelC3×Dic9C3×C18Dic9C3×C9C18C9C3×C6C32C6C6C3C3C2C1
# reps11222411323266

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

70
07
,
150
014
,
018
10
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

Export

Subgroup lattice of C3×Dic9 in TeX

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