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C5×#6878 #11925951

order1215
nilpotency class4
isoclinismisoclinic class of #6878
automorphism?
homologyAbelianization (H1) = C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×#6878

Subgroups

subgroupG/H
center, socleC3×C5#732
commutator, Frattini, cosocleC3×C9C32×C5
prime radicalC5×#732C3
LCSG ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1
UCSC1 ◁ C3×C5 ◁ C32×C5 ◁ C3×C5×C9 ◁ G

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 135

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C5×#6878, C5×#10624837, C5×#10624838, …
maximal normal subgroup ofC3×C5×#6878, C5×#10624837, C5×#10624838, …
maximal quotient ofC5×#10624481, C5×#10624483, C5×#10624488, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1215, 25);