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C5×#6859 #11925932

order1215
nilpotency class3
isoclinismisoclinic class of #6856
automorphism?
homologyAbelianization (H1) = C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×#6859

Subgroups

subgroupG/H
center, socleC32×C5H(3,𝔽3)
commutator, Frattini, cosocleC33C32×C5
prime radicalC3×C5×H(3,𝔽3)C3
LCSG ▷ C33 ▷ C32 ▷ C1
UCSC1 ◁ C32×C5 ◁ C33×C5 ◁ G

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 41
transitive degreeμt(G) = 405

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,3,4,5)(7,9,11)(15,16,19)(17,23,27)(18,22,28)(20,29,25)(21,32,26)(24,34,37)(30,40,41)(31,39,36)(33,38,35);(6,7,10)(8,11,14)(9,13,12)(15,17,24,26,36,41,18,25,35)(16,20,30,21,23,33,22,31,34)(19,27,37,32,39,40,28,29,38)

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C5×#6859, C5×#10624583, C5×#10624585, …
maximal normal subgroup ofC3×C5×#6859, C5×#10624583, C5×#10624585, …
maximal quotient ofC5×#10624449, C5×#10624450, C5×#10624451, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1215, 6);