Finite Group Databasebeta

← Prev (#11925927) Next (#11925929) →

C5×#6855 #11925928

order1215
nilpotency class2
isoclinismisoclinic class of H(3,𝔽3)
automorphism?
homologyAbelianization (H1) = C5×C92 · Schur multiplier (H2) = C33

Aliases

direct productC5×#6855

Subgroups

subgroupG/H
center, socle, prime radicalC33×C5C32
commutatorC3C5×C92
Frattini, cosocleC33C32×C5
LCSG ▷ C3 ▷ C1
UCSC1 ◁ C33×C5 ◁ G

138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 1215

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

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

Conjugacy classes & character table

495 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC5×#10624480, C5×#10624481, C5×#10624482, …
maximal normal subgroup ofC5×#10624480, C5×#10624481, C5×#10624482, …
maximal quotient ofC5×#10624463, C5×H(3,ℤ/9), C5×#10624465, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1215, 2);