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G = C3×D27order 162 = 2·34

Direct product of C3 and D27

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

Aliases: C3×D27, C273C6, C32.2D9, (C3×C27)⋊2C2, C9.2(C3×S3), (C3×C9).5S3, C3.2(C3×D9), SmallGroup(162,7)

Series: Derived Chief Lower central Upper central

C1C27 — C3×D27
C1C3C9C27C3×C27 — C3×D27
C27 — C3×D27
C1C3

Generators and relations for C3×D27
 G = < a,b,c | a3=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >

27C2
2C3
9S3
27C6
2C9
3D9
9C3×S3
2C27
3C3×D9

Smallest permutation representation of C3×D27
On 54 points
Generators in S54
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(28 37 46)(29 38 47)(30 39 48)(31 40 49)(32 41 50)(33 42 51)(34 43 52)(35 44 53)(36 45 54)
(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)
(1 32)(2 31)(3 30)(4 29)(5 28)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)

G:=sub<Sym(54)| (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,37,46)(29,38,47)(30,39,48)(31,40,49)(32,41,50)(33,42,51)(34,43,52)(35,44,53)(36,45,54), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)>;

G:=Group( (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,37,46)(29,38,47)(30,39,48)(31,40,49)(32,41,50)(33,42,51)(34,43,52)(35,44,53)(36,45,54), (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), (1,32)(2,31)(3,30)(4,29)(5,28)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33) );

G=PermutationGroup([(1,19,10),(2,20,11),(3,21,12),(4,22,13),(5,23,14),(6,24,15),(7,25,16),(8,26,17),(9,27,18),(28,37,46),(29,38,47),(30,39,48),(31,40,49),(32,41,50),(33,42,51),(34,43,52),(35,44,53),(36,45,54)], [(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)], [(1,32),(2,31),(3,30),(4,29),(5,28),(6,54),(7,53),(8,52),(9,51),(10,50),(11,49),(12,48),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,40),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33)])

C3×D27 is a maximal subgroup of   C273C18  C81⋊C6  C322D27  He3.3D9  He3.4D9  He3.5D9
C3×D27 is a maximal quotient of   C32⋊D27  C81⋊C6

45 conjugacy classes

class 1  2 3A3B3C3D3E6A6B9A···9I27A···27AA
order1233333669···927···27
size1271122227272···22···2

45 irreducible representations

dim1111222222
type+++++
imageC1C2C3C6S3C3×S3D9D27C3×D9C3×D27
kernelC3×D27C3×C27D27C27C3×C9C9C32C3C3C1
# reps11221239618

Matrix representation of C3×D27 in GL2(𝔽109) generated by

630
063
,
220
05
,
01
10
G:=sub<GL(2,GF(109))| [63,0,0,63],[22,0,0,5],[0,1,1,0] >;

C3×D27 in GAP, Magma, Sage, TeX

C_3\times D_{27}
% in TeX

G:=Group("C3xD27");
// GroupNames label

G:=SmallGroup(162,7);
// by ID

G=gap.SmallGroup(162,7);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,452,237,1803,138,2704]);
// Polycyclic

G:=Group<a,b,c|a^3=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D27 in TeX

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