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G = C27⋊C6order 162 = 2·34

The semidirect product of C27 and C6 acting faithfully

metacyclic, supersoluble, monomial

Aliases: C27⋊C6, D27⋊C3, C32.D9, C27⋊C3⋊C2, C9.3(C3×S3), (C3×C9).3S3, C3.3(C3×D9), SmallGroup(162,9)

Series: Derived Chief Lower central Upper central

C1C27 — C27⋊C6
C1C3C9C27C27⋊C3 — C27⋊C6
C27 — C27⋊C6
C1

Generators and relations for C27⋊C6
 G = < a,b | a27=b6=1, bab-1=a17 >

27C2
3C3
9S3
27C6
2C9
3D9
9C3×S3
2C27
3C3×D9

Character table of C27⋊C6

 class 123A3B3C6A6B9A9B9C9D9E27A27B27C27D27E27F27G27H27I
 size 127233272722266666666666
ρ1111111111111111111111    trivial
ρ21-1111-1-111111111111111    linear of order 2
ρ31-11ζ32ζ3ζ6ζ65111ζ3ζ321ζ3ζ3ζ32ζ32ζ32ζ311    linear of order 6
ρ41-11ζ3ζ32ζ65ζ6111ζ32ζ31ζ32ζ32ζ3ζ3ζ3ζ3211    linear of order 6
ρ5111ζ32ζ3ζ32ζ3111ζ3ζ321ζ3ζ3ζ32ζ32ζ32ζ311    linear of order 3
ρ6111ζ3ζ32ζ3ζ32111ζ32ζ31ζ32ζ32ζ3ζ3ζ3ζ3211    linear of order 3
ρ7202220022222-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ82022200-1-1-1-1-1ζ989ζ9792ζ9594ζ9792ζ9594ζ989ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ92022200-1-1-1-1-1ζ9792ζ9594ζ989ζ9594ζ989ζ9792ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ102022200-1-1-1-1-1ζ9594ζ989ζ9792ζ989ζ9792ζ9594ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ11202-1--3-1+-300222-1+-3-1--3-1ζ65ζ65ζ6ζ6ζ6ζ65-1-1    complex lifted from C3×S3
ρ12202-1+-3-1--300222-1--3-1+-3-1ζ6ζ6ζ65ζ65ζ65ζ6-1-1    complex lifted from C3×S3
ρ13202-1+-3-1--300-1-1-1ζ6ζ65ζ9594ζ9795ζ9894ζ9492ζ959ζ9897ζ929ζ989ζ9792    complex lifted from C3×D9
ρ14202-1+-3-1--300-1-1-1ζ6ζ65ζ9792ζ929ζ9795ζ9897ζ9492ζ959ζ9894ζ9594ζ989    complex lifted from C3×D9
ρ15202-1--3-1+-300-1-1-1ζ65ζ6ζ9792ζ9897ζ9492ζ929ζ9795ζ9894ζ959ζ9594ζ989    complex lifted from C3×D9
ρ16202-1--3-1+-300-1-1-1ζ65ζ6ζ9594ζ9492ζ959ζ9795ζ9894ζ929ζ9897ζ989ζ9792    complex lifted from C3×D9
ρ17202-1--3-1+-300-1-1-1ζ65ζ6ζ989ζ959ζ9897ζ9894ζ929ζ9795ζ9492ζ9792ζ9594    complex lifted from C3×D9
ρ18202-1+-3-1--300-1-1-1ζ6ζ65ζ989ζ9894ζ929ζ959ζ9897ζ9492ζ9795ζ9792ζ9594    complex lifted from C3×D9
ρ1960-3000097+3ζ9295+3ζ9498+3ζ900000000000    orthogonal faithful
ρ2060-3000095+3ζ9498+3ζ997+3ζ9200000000000    orthogonal faithful
ρ2160-3000098+3ζ997+3ζ9295+3ζ9400000000000    orthogonal faithful

Permutation representations of C27⋊C6
On 27 points - transitive group 27T55
Generators in S27
(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)
(2 9 11 27 20 18)(3 17 21 26 12 8)(4 25)(5 6 14 24 23 15)(7 22)(10 19)(13 16)

G:=sub<Sym(27)| (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), (2,9,11,27,20,18)(3,17,21,26,12,8)(4,25)(5,6,14,24,23,15)(7,22)(10,19)(13,16)>;

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), (2,9,11,27,20,18)(3,17,21,26,12,8)(4,25)(5,6,14,24,23,15)(7,22)(10,19)(13,16) );

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)], [(2,9,11,27,20,18),(3,17,21,26,12,8),(4,25),(5,6,14,24,23,15),(7,22),(10,19),(13,16)])

G:=TransitiveGroup(27,55);

C27⋊C6 is a maximal subgroup of   C27⋊C18  C33.D9  He3.3D9  He3.4D9  C33.5D9  He3.5D9
C27⋊C6 is a maximal quotient of   C27⋊C12  C273C18  C32⋊D27  C33.5D9

Matrix representation of C27⋊C6 in GL6(𝔽109)

00598200
00273200
00005982
00002732
100000
010000
,
010000
100000
00003250
00008277
00502700
00775900

G:=sub<GL(6,GF(109))| [0,0,0,0,1,0,0,0,0,0,0,1,59,27,0,0,0,0,82,32,0,0,0,0,0,0,59,27,0,0,0,0,82,32,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,50,77,0,0,0,0,27,59,0,0,32,82,0,0,0,0,50,77,0,0] >;

C27⋊C6 in GAP, Magma, Sage, TeX

C_{27}\rtimes C_6
% in TeX

G:=Group("C27:C6");
// GroupNames label

G:=SmallGroup(162,9);
// by ID

G=gap.SmallGroup(162,9);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,452,457,237,1803,138,2704]);
// Polycyclic

G:=Group<a,b|a^27=b^6=1,b*a*b^-1=a^17>;
// generators/relations

Export

Subgroup lattice of C27⋊C6 in TeX
Character table of C27⋊C6 in TeX

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