Finite Group Databasebeta

← Prev (#13062895) Next (#13062897) →

C5×#63270 #13062896

order1365
isoclinismisoclinic class of #63270
automorphism?
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×#63270

Splittings

Fitting, socle, HallC5×C7×C13 ⋊ C3
derived, HallC7×C13 ⋊ C3×C5
HallC5×C13 ⋊ ΓL(1,𝔽8)
HallC5×C7 ⋊ C13⋊C3
SylowC13 ⋊ C5×ΓL(1,𝔽8)
SylowC7 ⋊ C5×C13⋊C3

Subgroups

subgroupG/H
center, hypercenterC5#63270
commutator, cosocle, hypocenterC7×C13C3×C5
FrattiniC1G
Fitting, socleC5×C7×C13C3
LCSG ▷ C7×C13
UCSC1 ◁ C5

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 455

A minimal faithful permutation representation — 2 generators on 25 points:

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

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C5×#63270, C5×#10587716, C5×#10587720, …
maximal normal subgroup ofC2×C5×#63270, C5×#10587716, C5×#10587720, …
maximal quotient ofC2×C5×#63270, C5×#11717941, C3×C5×#63270, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1365, 4);