Copied to
clipboard

## G = S3×D27order 324 = 22·34

### Direct product of S3 and D27

Aliases: S3×D27, C271D6, C31D54, C32.2D18, C27⋊S3⋊C2, C9.1S32, (S3×C27)⋊C2, (C3×D27)⋊C2, (S3×C9).S3, (C3×C27)⋊C22, (C3×S3).D9, C3.2(S3×D9), (C3×C9).5D6, SmallGroup(324,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C27 — S3×D27
 Chief series C1 — C3 — C9 — C3×C9 — C3×C27 — S3×C27 — S3×D27
 Lower central C3×C27 — S3×D27
 Upper central C1

Generators and relations for S3×D27
G = < a,b,c,d | a3=b2=c27=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

3C2
27C2
81C2
2C3
81C22
3C6
9S3
27S3
27S3
27C6
54S3
2C9
27D6
27D6
3C18
3D9
9D9
18D9
2C27
9D18
9S32
3D27
3C54
6D27
3D54

Smallest permutation representation of S3×D27
On 54 points
Generators in S54
(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 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(23 34)(24 35)(25 36)(26 37)(27 38)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 54)(14 53)(15 52)(16 51)(17 50)(18 49)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 42)(26 41)(27 40)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 6A 6B 9A 9B 9C 9D 9E 9F 18A 18B 18C 27A ··· 27I 27J ··· 27R 54A ··· 54I order 1 2 2 2 3 3 3 6 6 9 9 9 9 9 9 18 18 18 27 ··· 27 27 ··· 27 54 ··· 54 size 1 3 27 81 2 2 4 6 54 2 2 2 4 4 4 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 S3 D6 D6 D9 D18 D27 D54 S32 S3×D9 S3×D27 kernel S3×D27 C3×D27 S3×C27 C27⋊S3 D27 S3×C9 C27 C3×C9 C3×S3 C32 S3 C3 C9 C3 C1 # reps 1 1 1 1 1 1 1 1 3 3 9 9 1 3 9

Matrix representation of S3×D27 in GL4(𝔽109) generated by

 1 0 0 0 0 1 0 0 0 0 108 1 0 0 108 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 7 63 0 0 75 99 0 0 0 0 1 0 0 0 0 1
,
 108 0 0 0 2 1 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(109))| [1,0,0,0,0,1,0,0,0,0,108,108,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[7,75,0,0,63,99,0,0,0,0,1,0,0,0,0,1],[108,2,0,0,0,1,0,0,0,0,1,0,0,0,0,1] >;

S3×D27 in GAP, Magma, Sage, TeX

S_3\times D_{27}
% in TeX

G:=Group("S3xD27");
// GroupNames label

G:=SmallGroup(324,38);
// by ID

G=gap.SmallGroup(324,38);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,404,824,579,2710,208,3899]);
// Polycyclic

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

Export

׿
×
𝔽