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Cayley diagram

D5×H(3,𝔽3) #63188

order270
isoclinismisoclinic class of D5×H(3,𝔽3)
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32

Aliases

direct productD5×H(3,𝔽3)
unambiguous semidirect productC5⋊(C2×H(3,𝔽3))
unambiguous semidirect product(C32×D5)⋊C3

Splittings

Fitting, HallC5×H(3,𝔽3) ⋊ C2
SylowC5 ⋊ C2×H(3,𝔽3)

Subgroups

subgroupG/H
center, FrattiniC3C32×D5
commutator, socle, cosocleC3×C5C2×C32
FittingC5×H(3,𝔽3)C2
hypercenterH(3,𝔽3)D5
hypocenterC5C2×H(3,𝔽3)
LCSG ▷ C3×C5 ▷ C5
UCSC1 ◁ C3 ◁ H(3,𝔽3)

44 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)(6,7,9)(8,11,13)(10,14,12)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×H(3,𝔽3), (C3⋊ΓL(1,𝔽16))⋊C3, #10587380, …
maximal normal subgroup ofAGL(1,𝔽5)×H(3,𝔽3), (C3⋊ΓL(1,𝔽16))⋊C3, #10587380, …
maximal quotient ofDic5×H(3,𝔽3), C2×D5×H(3,𝔽3), D5×C32⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(270, 6);