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G = S3×D27order 324 = 22·34

Direct product of S3 and D27

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

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

C1C3×C27 — S3×D27
C1C3C9C3×C9C3×C27S3×C27 — S3×D27
C3×C27 — S3×D27
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
9C3⋊S3
9C3×S3
18D9
2C27
9D18
9S32
3C3×D9
3D27
3C9⋊S3
3C54
6D27
3D54
3S3×D9

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

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

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

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

45 conjugacy classes

class 1 2A2B2C3A3B3C6A6B9A9B9C9D9E9F18A18B18C27A···27I27J···27R54A···54I
order12223336699999918181827···2727···2754···54
size1327812246542224446662···24···46···6

45 irreducible representations

dim111122222222444
type+++++++++++++++
imageC1C2C2C2S3S3D6D6D9D18D27D54S32S3×D9S3×D27
kernelS3×D27C3×D27S3×C27C27⋊S3D27S3×C9C27C3×C9C3×S3C32S3C3C9C3C1
# reps111111113399139

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

1000
0100
001081
001080
,
1000
0100
0001
0010
,
76300
759900
0010
0001
,
108000
2100
0010
0001
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

Subgroup lattice of S3×D27 in TeX

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