Finite Group Databasebeta

← Prev (#63183) Next (#63185) →

Cayley diagram

D5×C27 #63184

order270
isoclinismisoclinic class of D5
automorphismAut(G) = C2×C9×AGL(1,𝔽5) (order 360) · Inn(G) = D5 · Out(G) = C22×C9
homologyAbelianization (H1) = C2×C27 · Schur multiplier (H2) = C1

Aliases

direct productD5×C27
unambiguous semidirect productC5⋊(C2×C27)

Splittings

Fitting, HallC5×C27 ⋊ C2
derived, SylowC5 ⋊ C2×C27

Subgroups

subgroupG/H
center, hypercenterC27D5
commutator, hypocenterC5C2×C27
FrattiniC9C3×D5
FittingC5×C27C2
socleC3×C5C2×C9
prime radicalC3×D5C9
cosocleC5×C9C2×C3
LCSG ▷ C5
UCSC1 ◁ C27

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:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×C27, C27⋊AGL(1,𝔽5), D5×D27, …
maximal normal subgroup ofAGL(1,𝔽5)×C27, C27⋊AGL(1,𝔽5), D5×D27, …
maximal quotient ofDic5×C27, C2×D5×C27, D5×C81, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 2);