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G = C9×D27order 486 = 2·35

Direct product of C9 and D27

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

Aliases: C9×D27, C275C18, C92.3S3, (C9×C27)⋊2C2, C9.3(S3×C9), C3.2(C9×D9), (C3×C9).6D9, (C3×C27).6C6, C3.4(C3×D27), (C3×D27).2C3, C32.12(C3×D9), (C3×C9).52(C3×S3), SmallGroup(486,13)

Series: Derived Chief Lower central Upper central

C1C27 — C9×D27
C1C3C9C27C3×C27C9×C27 — C9×D27
C27 — C9×D27
C1C9

Generators and relations for C9×D27
 G = < a,b,c | a9=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >

27C2
2C3
9S3
27C6
2C9
2C9
2C9
2C9
2C9
3D9
9C3×S3
27C18
2C27
2C27
2C3×C9
2C27
2C27
3C3×D9
9S3×C9
2C3×C27
3C9×D9

Smallest permutation representation of C9×D27
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 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)

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

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

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

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+++++
imageC1C2C3C6C9C18S3D9C3×S3D27S3×C9C3×D9C9×D9C3×D27C9×D27
kernelC9×D27C9×C27C3×D27C3×C27D27C27C92C3×C9C3×C9C9C9C32C3C3C1
# reps112266132966181854

Matrix representation of C9×D27 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] >;

C9×D27 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 C9×D27 in TeX

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