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## G = S3×C27order 162 = 2·34

### Direct product of C27 and S3

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

 Derived series C1 — C3 — S3×C27
 Chief series C1 — C3 — C32 — C3×C9 — C3×C27 — S3×C27
 Lower central C3 — S3×C27
 Upper central C1 — C27

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

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

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

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

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

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 3A 3B 3C 3D 3E 6A 6B 9A ··· 9F 9G ··· 9L 18A ··· 18F 27A ··· 27R 27S ··· 27AJ 54A ··· 54R order 1 2 3 3 3 3 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 27 ··· 27 27 ··· 27 54 ··· 54 size 1 3 1 1 2 2 2 3 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C6 C9 C18 C27 C54 S3 C3×S3 S3×C9 S3×C27 kernel S3×C27 C3×C27 S3×C9 C3×C9 C3×S3 C32 S3 C3 C27 C9 C3 C1 # reps 1 1 2 2 6 6 18 18 1 2 6 18

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

 5 0 0 5
,
 63 0 45 45
,
 108 44 0 1
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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