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C5×#6912 #11925985

order1215
nilpotency class3
isoclinismisoclinic class of #6909
automorphism?
homologyAbelianization (H1) = C33×C5 · Schur multiplier (H2) = C33

Aliases

direct productC5×#6912

Subgroups

subgroupG/H
center, socleC3×C5C3×H(3,𝔽3)
commutator, Frattini, cosocleC32C33×C5
prime radicalC33×C5C32
LCSG ▷ C32 ▷ C3 ▷ C1
UCSC1 ◁ C3×C5 ◁ C33×C5 ◁ G

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×#10624750, C5×#10624751, C5×#10624752, …
maximal normal subgroup ofC5×#10624750, C5×#10624751, C5×#10624752, …
maximal quotient ofC5×#10624592, C5×#10624594, C5×#10624597, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1215, 59);