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G = S3×C27order 162 = 2·34

Direct product of C27 and S3

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

Aliases: S3×C27, C3⋊C54, C32.2C18, (C3×C27)⋊1C2, (S3×C9).C3, (C3×S3).C9, C9.4(C3×S3), C3.4(S3×C9), (C3×C9).5C6, SmallGroup(162,8)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C27
C1C3C32C3×C9C3×C27 — S3×C27
C3 — S3×C27
C1C27

Generators and relations for S3×C27
 G = < a,b,c | a27=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
2C3
3C6
2C9
3C18
2C27
3C54

Smallest permutation representation of S3×C27
On 54 points
Generators in S54
(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 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 28)(24 29)(25 30)(26 31)(27 32)

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

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

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

S3×C27 is a maximal subgroup of   C32⋊C54  He3.C18  He3.2C18  C9⋊C54  He3.5C18
S3×C27 is a maximal quotient of   C32⋊C54  C9⋊C54

81 conjugacy classes

class 1  2 3A3B3C3D3E6A6B9A···9F9G···9L18A···18F27A···27R27S···27AJ54A···54R
order1233333669···99···918···1827···2727···2754···54
size1311222331···12···23···31···12···23···3

81 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C9C18C27C54S3C3×S3S3×C9S3×C27
kernelS3×C27C3×C27S3×C9C3×C9C3×S3C32S3C3C27C9C3C1
# reps112266181812618

Matrix representation of S3×C27 in GL2(𝔽109) generated by

50
05
,
630
4545
,
10844
01
G:=sub<GL(2,GF(109))| [5,0,0,5],[63,45,0,45],[108,0,44,1] >;

S3×C27 in GAP, Magma, Sage, TeX

S_3\times C_{27}
% in TeX

G:=Group("S3xC27");
// GroupNames label

G:=SmallGroup(162,8);
// by ID

G=gap.SmallGroup(162,8);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,36,57,2704]);
// Polycyclic

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

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Subgroup lattice of S3×C27 in TeX

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