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Cayley diagram

D27 #265

order54
isoclinismisoclinic class of D27
automorphismAut(G) = AGL(1,ℤ/27) (order 486) · Inn(G) = G · Out(G) = C9
holomorphG ⋊ Aut(G) = #72214444
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD27
unambiguous semidirect productC27⋊C2

Splittings

Fitting, derived, SylowC27 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC27C2
FrattiniC9D3
socleC3D9
LCSG ▷ C27
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 27

A minimal faithful permutation representation — 2 generators on 27 points:

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

Polycyclic presentation:

2,3^3|b^3=c^2d,c^3=d^2,b^a=b^2c,c^a=c^2d,d^a=d^2

Conjugacy classes & character table

15 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D27, D81, C3×D27, …
maximal normal subgroup ofC2×D27, C3×D27, D27⋊C3, …
maximal quotient ofDic27, C2×D27, D81, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(54, 1);