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Cayley diagram

D5×3−1+2 #63189

order270
isoclinismisoclinic class of D5×H(3,𝔽3)
automorphismAut(G) = AGL(1,𝔽5)×#272 (order 1080) · Inn(G) = C32×D5 · Out(G) = C2×D3
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C1

Aliases

direct productD5×3−1+2
unambiguous semidirect productC5⋊(C2×3−1+2)
unambiguous semidirect product(C9×D5)⋊C3

Splittings

Fitting, HallC5×3−1+2 ⋊ C2
SylowC5 ⋊ C2×3−1+2

Subgroups

subgroupG/H
center, FrattiniC3C32×D5
commutator, socle, cosocleC3×C5C2×C32
FittingC5×3−1+2C2
prime radicalC32×D5C3
hypercenter3−1+2D5
hypocenterC5C2×3−1+2
LCSG ▷ C3×C5 ▷ C5
UCSC1 ◁ C3 ◁ 3−1+2

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 45

A minimal faithful permutation representation — 2 generators on 14 points:

(2,3)(4,5)(7,10,11)(9,13,14);(1,2,4,5,3)(6,11,14,8,10,9,12,7,13)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×3−1+2, 3−1+2⋊AGL(1,𝔽5), D5×AGL(1,ℤ/9), …
maximal normal subgroup ofAGL(1,𝔽5)×3−1+2, 3−1+2⋊AGL(1,𝔽5), D5×AGL(1,ℤ/9), …
maximal quotient ofDic5×3−1+2, C2×D5×3−1+2, D5×C32⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 7);