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C3×C5×#731 #11925978

order1215
nilpotency class3
isoclinismisoclinic class of AΣL(1,𝔽27)
automorphism?
homologyAbelianization (H1) = C33×C5 · Schur multiplier (H2) = C33

Aliases

direct productC3×C5×#731

Subgroups

subgroupG/H
center, socleC32×C5H(3,𝔽3)
commutator, Frattini, cosocleC32C33×C5
prime radicalC3×C5×H(3,𝔽3)C3
LCSG ▷ C32 ▷ C3 ▷ C1
UCSC1 ◁ C32×C5 ◁ C33×C5 ◁ G

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 405

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(31,32,33,34,35);(4,5,8)(6,12,16)(7,11,17)(9,18,14)(10,21,15)(13,23,26)(19,29,30)(20,28,25)(22,27,24);(4,6,13,7,14,24,15,25,30)(5,9,19,11,20,23,10,12,22)(8,16,26,17,18,27,21,28,29)

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×C9×#731, C5×#731⋊C9, C5×#10624731, …
maximal normal subgroup ofC5×C9×#731, C5×#731⋊C9, C5×#10624731, …
maximal quotient ofC3×C5×#6857, C3×C5×#6858, C3×C5×#6859, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1215, 52);