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G = C81⋊C6  order 486 = 2·35

The semidirect product of C81 and C6 acting faithfully

metacyclic, supersoluble, monomial

Aliases: C81⋊C6, D81⋊C3, C32.D27, C81⋊C3⋊C2, (C3×C9).4D9, C9.3(C3×D9), C27.2(C3×S3), (C3×C27).2S3, C3.3(C3×D27), SmallGroup(486,34)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C81 — C81⋊C6
Chief series
C1 — C3 — C9 — C27 — C81 — C81⋊C3 — C81⋊C6
Lower central
C81 — C81⋊C6
Upper central
C1

Generators and relations for C81⋊C6
 G = < a,b | a81=b6=1, bab-1=a26 >

C1
81C2
C3
3C3
27S3
81C6
C9
C32
2C9
9D9
27C3×S3
C27
C3×C9
2C27
3D27
9C3×D9
C3×C27
C81
2C81
D81
3C3×D27
C81⋊C3
C81⋊C6

Smallest permutation representation of C81⋊C6
►On 81 points
Generators in S81
(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 54 56 81 29 27)(3 26 30 80 57 53)(4 79)(5 51 59 78 32 24)(6 23 33 77 60 50)(7 76)(8 48 62 75 35 21)(9 20 36 74 63 47)(10 73)(11 45 65 72 38 18)(12 17 39 71 66 44)(13 70)(14 42 68 69 41 15)(16 67)(19 64)(22 61)(25 58)(28 55)(31 52)(34 49)(37 46)(40 43)
 
G:=sub<Sym(81)| (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), (2,54,56,81,29,27)(3,26,30,80,57,53)(4,79)(5,51,59,78,32,24)(6,23,33,77,60,50)(7,76)(8,48,62,75,35,21)(9,20,36,74,63,47)(10,73)(11,45,65,72,38,18)(12,17,39,71,66,44)(13,70)(14,42,68,69,41,15)(16,67)(19,64)(22,61)(25,58)(28,55)(31,52)(34,49)(37,46)(40,43)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81), (2,54,56,81,29,27)(3,26,30,80,57,53)(4,79)(5,51,59,78,32,24)(6,23,33,77,60,50)(7,76)(8,48,62,75,35,21)(9,20,36,74,63,47)(10,73)(11,45,65,72,38,18)(12,17,39,71,66,44)(13,70)(14,42,68,69,41,15)(16,67)(19,64)(22,61)(25,58)(28,55)(31,52)(34,49)(37,46)(40,43) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81)], [(2,54,56,81,29,27),(3,26,30,80,57,53),(4,79),(5,51,59,78,32,24),(6,23,33,77,60,50),(7,76),(8,48,62,75,35,21),(9,20,36,74,63,47),(10,73),(11,45,65,72,38,18),(12,17,39,71,66,44),(13,70),(14,42,68,69,41,15),(16,67),(19,64),(22,61),(25,58),(28,55),(31,52),(34,49),(37,46),(40,43)]])
 

54 conjugacy classes

class 1  2 3A3B3C6A6B9A9B9C9D9E27A···27I27J···27O81A···81AA
order12333669999927···2727···2781···81
size1812338181222662···26···66···6

54 irreducible representations

dim11112222226
type++++++
imageC1C2C3C6S3C3×S3D9C3×D9D27C3×D27C81⋊C6
kernelC81⋊C6C81⋊C3D81C81C3×C27C27C3×C9C9C32C3C1
# reps112212369189

Matrix representation of C81⋊C6 ►in GL6(𝔽163)

001488300
00806800
107871081081265
1108767611012
2375904413176
117121571086411
,
119180000
137440000
1591368814215
126911241241536
1317112114478
2512811910069147

G:=sub<GL(6,GF(163))| [0,0,107,1,23,117,0,0,87,108,75,12,148,80,108,76,90,157,83,68,108,76,44,108,0,0,12,110,131,64,0,0,65,12,76,11],[119,137,159,126,13,25,18,44,136,91,17,128,0,0,8,124,112,119,0,0,8,124,114,100,0,0,142,15,47,69,0,0,15,36,8,147] >;
 

C81⋊C6 in GAP, Magma, Sage, TeX 

C_{81}\rtimes C_6
 
% in TeX
 
G:=Group("C81:C6");
 
// GroupNames label
 
G:=SmallGroup(486,34);
 
// by ID
 
G=gap.SmallGroup(486,34);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,542,1520,284,2163,381,8104,208,11669]);
 
// Polycyclic
 
G:=Group<a,b|a^81=b^6=1,b*a*b^-1=a^26>;
 
// generators/relations
 

Export

Subgroup lattice of C81⋊C6 in TeX

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