Copied to
clipboard

G = C9xD27  order 486 = 2·35

Direct product of C9 and D27

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

Aliases: C9xD27, C27:5C18, C92.3S3, (C9xC27):2C2, C9.3(S3xC9), C3.2(C9xD9), (C3xC9).6D9, (C3xC27).6C6, C3.4(C3xD27), (C3xD27).2C3, C32.12(C3xD9), (C3xC9).52(C3xS3), SmallGroup(486,13)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C27 — C9xD27
Chief series
C1 — C3 — C9 — C27 — C3xC27 — C9xC27 — C9xD27
Lower central
C27 — C9xD27
Upper central
C1 — C9

Generators and relations for C9xD27
 G = < a,b,c | a9=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 156 in 36 conjugacy classes, 15 normal (all characteristic)
Quotients: C1, C2, C3, S3, C6, C9, D9, C18, C3xS3, D27, C3xD9, S3xC9, C9xD9, C3xD27, C9xD27
C1
27C2
C3
C3
2C3
9S3
27C6
C9
C9
C32
2C9
2C9
2C9
2C9
2C9
3D9
9C3xS3
27C18
C3xC9
C27
C3xC9
2C27
2C27
2C3xC9
2C27
2C27
D27
3C3xD9
9S3xC9
C92
C3xC27
2C3xC27
C3xD27
3C9xD9
C9xC27
C9xD27

Smallest permutation representation of C9xD27
►On 54 points
Generators in S54
(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)(28 52 49 46 43 40 37 34 31)(29 53 50 47 44 41 38 35 32)(30 54 51 48 45 42 39 36 33)

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

135 conjugacy classes

class 1  2 3A3B3C3D3E6A6B9A···9F9G···9AM18A···18F27A···27CC
order1233333669···99···918···1827···27
size1271122227271···12···227···272···2

135 irreducible representations

dim111111222222222
type+++++
imageC1C2C3C6C9C18S3D9C3xS3D27S3xC9C3xD9C9xD9C3xD27C9xD27
kernelC9xD27C9xC27C3xD27C3xC27D27C27C92C3xC9C3xC9C9C9C32C3C3C1
# reps112266132966181854

Matrix representation of C9xD27 ►in GL2(F109) generated by

160
016
,
534
022
,
8899
4421
G:=sub<GL(2,GF(109))| [16,0,0,16],[5,0,34,22],[88,44,99,21] >;
 

C9xD27 in GAP, Magma, Sage, TeX 

C_9\times D_{27}
 
% in TeX
 
G:=Group("C9xD27");
 
// GroupNames label
 
G:=SmallGroup(486,13);
 
// by ID
 
G=gap.SmallGroup(486,13);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,2163,381,8104,208,11669]);
 
// Polycyclic
 
G:=Group<a,b,c|a^9=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C9xD27 in TeX

׿
x
:
Z
F
o
wr
Q
<