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Cayley diagram

C5×D27 #63183

order270
isoclinismisoclinic class of D27
automorphismAut(G) = C4×AGL(1,ℤ/27) (order 1944) · Inn(G) = D27 · Out(G) = C4×C9
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D27
unambiguous semidirect productC27⋊(C2×C5)

Splittings

Fitting, HallC5×C27 ⋊ C2
derived, SylowC27 ⋊ C2×C5

Subgroups

subgroupG/H
center, hypercenterC5D27
commutator, cosocle, hypocenterC27C2×C5
FrattiniC9C5×D3
FittingC5×C27C2
socleC3×C5D9
LCSG ▷ C27
UCSC1 ◁ C5

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 135

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,3,4,5)(7,10)(8,14)(9,17)(11,12)(13,23)(15,16)(18,25)(19,24)(20,30)(21,29)(22,28)(26,27)(31,32);(6,25,19,26,30,28,8,7,29,9,32,23,15,12,11,16,13,31,17,21,10,14,22,20,27,24,18)

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×D27, C2×C5×D27, C5×D81, …
maximal normal subgroup ofD5×D27, C2×C5×D27, C3×C5×D27, …
maximal quotient ofC5×Dic27, C2×C5×D27, C5×D81, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 1);