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G = C32⋊D27  order 486 = 2·35

1st semidirect product of C32 and D27 acting via D27/C9=S3

metabelian, supersoluble, monomial

Aliases: C32⋊1D27, C33.1D9, C27⋊S3⋊1C3, C3.(C27⋊C6), (C3×C27)⋊1C6, C9.4(C9⋊C6), C32⋊C27⋊2C2, C3.1(C3×D27), (C32×C9).8S3, C9.3(C32⋊C6), C32.14(C3×D9), C3.1(C32⋊D9), (C3×C9).54(C3×S3), SmallGroup(486,17)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C27 — C32⋊D27
Chief series
C1 — C3 — C9 — C3×C9 — C3×C27 — C32⋊C27 — C32⋊D27
Lower central
C3×C27 — C32⋊D27
Upper central
C1

Generators and relations for C32⋊D27
 G = < a,b,c,d | a3=b3=c27=d2=1, cac-1=ab=ba, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

C1
81C2
C3
C3
C3
C3
3C3
6C3
27S3
27S3
27S3
27S3
81C6
C32
C9
C32
C9
C9
2C32
3C32
3C32
3C32
6C9
9D9
9C3⋊S3
9D9
9D9
27C3×S3
27C3×S3
27C3×S3
27C3×S3
C3×C9
C33
2C3×C9
3C3×C9
3C3×C9
3C27
3C3×C9
6C27
3C9⋊S3
9C3×C3⋊S3
9C3×D9
9D27
9C3×D9
9C3×D9
C3×C27
C32×C9
2C3×C27
C27⋊S3
3C3×C9⋊S3
C32⋊C27
C32⋊D27

Smallest permutation representation of C32⋊D27
►On 81 points
Generators in S81
(2 79 39)(3 40 80)(5 55 42)(6 43 56)(8 58 45)(9 46 59)(11 61 48)(12 49 62)(14 64 51)(15 52 65)(17 67 54)(18 28 68)(20 70 30)(21 31 71)(23 73 33)(24 34 74)(26 76 36)(27 37 77)

(1 78 38)(2 79 39)(3 80 40)(4 81 41)(5 55 42)(6 56 43)(7 57 44)(8 58 45)(9 59 46)(10 60 47)(11 61 48)(12 62 49)(13 63 50)(14 64 51)(15 65 52)(16 66 53)(17 67 54)(18 68 28)(19 69 29)(20 70 30)(21 71 31)(22 72 32)(23 73 33)(24 74 34)(25 75 35)(26 76 36)(27 77 37)

(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)(55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81)

(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 81)(36 80)(37 79)(38 78)(39 77)(40 76)(41 75)(42 74)(43 73)(44 72)(45 71)(46 70)(47 69)(48 68)(49 67)(50 66)(51 65)(52 64)(53 63)(54 62)
 
G:=sub<Sym(81)| (2,79,39)(3,40,80)(5,55,42)(6,43,56)(8,58,45)(9,46,59)(11,61,48)(12,49,62)(14,64,51)(15,52,65)(17,67,54)(18,28,68)(20,70,30)(21,31,71)(23,73,33)(24,34,74)(26,76,36)(27,37,77), (1,78,38)(2,79,39)(3,80,40)(4,81,41)(5,55,42)(6,56,43)(7,57,44)(8,58,45)(9,59,46)(10,60,47)(11,61,48)(12,62,49)(13,63,50)(14,64,51)(15,65,52)(16,66,53)(17,67,54)(18,68,28)(19,69,29)(20,70,30)(21,71,31)(22,72,32)(23,73,33)(24,74,34)(25,75,35)(26,76,36)(27,77,37), (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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,81)(36,80)(37,79)(38,78)(39,77)(40,76)(41,75)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62)>;
 
G:=Group( (2,79,39)(3,40,80)(5,55,42)(6,43,56)(8,58,45)(9,46,59)(11,61,48)(12,49,62)(14,64,51)(15,52,65)(17,67,54)(18,28,68)(20,70,30)(21,31,71)(23,73,33)(24,34,74)(26,76,36)(27,37,77), (1,78,38)(2,79,39)(3,80,40)(4,81,41)(5,55,42)(6,56,43)(7,57,44)(8,58,45)(9,59,46)(10,60,47)(11,61,48)(12,62,49)(13,63,50)(14,64,51)(15,65,52)(16,66,53)(17,67,54)(18,68,28)(19,69,29)(20,70,30)(21,71,31)(22,72,32)(23,73,33)(24,74,34)(25,75,35)(26,76,36)(27,77,37), (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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,81)(36,80)(37,79)(38,78)(39,77)(40,76)(41,75)(42,74)(43,73)(44,72)(45,71)(46,70)(47,69)(48,68)(49,67)(50,66)(51,65)(52,64)(53,63)(54,62) );
 
G=PermutationGroup([[(2,79,39),(3,40,80),(5,55,42),(6,43,56),(8,58,45),(9,46,59),(11,61,48),(12,49,62),(14,64,51),(15,52,65),(17,67,54),(18,28,68),(20,70,30),(21,31,71),(23,73,33),(24,34,74),(26,76,36),(27,37,77)], [(1,78,38),(2,79,39),(3,80,40),(4,81,41),(5,55,42),(6,56,43),(7,57,44),(8,58,45),(9,59,46),(10,60,47),(11,61,48),(12,62,49),(13,63,50),(14,64,51),(15,65,52),(16,66,53),(17,67,54),(18,68,28),(19,69,29),(20,70,30),(21,71,31),(22,72,32),(23,73,33),(24,74,34),(25,75,35),(26,76,36),(27,77,37)], [(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),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)], [(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(28,61),(29,60),(30,59),(31,58),(32,57),(33,56),(34,55),(35,81),(36,80),(37,79),(38,78),(39,77),(40,76),(41,75),(42,74),(43,73),(44,72),(45,71),(46,70),(47,69),(48,68),(49,67),(50,66),(51,65),(52,64),(53,63),(54,62)]])
 

54 conjugacy classes

class 1  2 3A3B3C3D3E3F3G3H6A6B9A···9I9J···9O27A···27AA
order1233333333669···99···927···27
size1812222336681812···26···66···6

54 irreducible representations

dim1111222222666
type++++++++
imageC1C2C3C6S3C3×S3D9D27C3×D9C3×D27C32⋊C6C9⋊C6C27⋊C6
kernelC32⋊D27C32⋊C27C27⋊S3C3×C27C32×C9C3×C9C33C32C32C3C9C9C3
# reps11221239618126

Matrix representation of C32⋊D27 ►in GL8(𝔽109)

450000000
045000000
00100000
00010000
00512210810800
0087291000
0079630001
00461600108108
,
10000000
01000000
00010000
001081080000
00000100
000010810800
00000001
000000108108
,
6379000000
3093000000
001010223500
007171041800
00009975927
0000102928232
00009210200
000079900
,
5932000000
8250000000
0032820000
0050770000
0000002777
0000005082
0000277700
0000508200

G:=sub<GL(8,GF(109))| [45,0,0,0,0,0,0,0,0,45,0,0,0,0,0,0,0,0,1,0,51,87,79,46,0,0,0,1,22,29,63,16,0,0,0,0,108,1,0,0,0,0,0,0,108,0,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,1,108],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,1,108,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,1,108,0,0,0,0,0,0,0,0,0,108,0,0,0,0,0,0,1,108],[63,30,0,0,0,0,0,0,79,93,0,0,0,0,0,0,0,0,10,7,0,0,0,0,0,0,102,17,0,0,0,0,0,0,23,104,99,102,92,7,0,0,5,18,7,92,102,99,0,0,0,0,59,82,0,0,0,0,0,0,27,32,0,0],[59,82,0,0,0,0,0,0,32,50,0,0,0,0,0,0,0,0,32,50,0,0,0,0,0,0,82,77,0,0,0,0,0,0,0,0,0,0,27,50,0,0,0,0,0,0,77,82,0,0,0,0,27,50,0,0,0,0,0,0,77,82,0,0] >;
 

C32⋊D27 in GAP, Magma, Sage, TeX 

C_3^2\rtimes D_{27}
 
% in TeX
 
G:=Group("C3^2:D27");
 
// GroupNames label
 
G:=SmallGroup(486,17);
 
// by ID
 
G=gap.SmallGroup(486,17);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,224,824,867,8104,208,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^3=c^27=d^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C32⋊D27 in TeX

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