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C5×#6866 #11925939

order1215
nilpotency class3
isoclinismisoclinic class of AΣL(1,𝔽27)
automorphism?
homologyAbelianization (H1) = C3×C5×C9 · Schur multiplier (H2) = C32

Aliases

direct productC5×#6866

Subgroups

subgroupG/H
center, socleC32×C5H(3,𝔽3)
commutatorC32C3×C5×C9
Frattini, cosocleC33C32×C5
prime radicalC34×C5C3
LCSG ▷ C32 ▷ C3 ▷ C1
UCSC1 ◁ C32×C5 ◁ C33×C5 ◁ G

180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 135

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,3,4,5)(6,7,9,8,10,12,11,13,14)(15,16,18)(17,19,22)(20,23,21);(16,19,20)

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×#10624474, C5×#10624475, C5×#10624476, …
maximal normal subgroup ofC5×#10624474, C5×#10624475, C5×#10624476, …
maximal quotient ofC5×#10624444, C5×#10624449, C5×#10624452, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1215, 13);