Finite Group Databasebeta

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Cayley diagram

C5×#1044 #92558

order500
isoclinismisoclinic class of #1044
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C5

Aliases

direct productC5×#1044

Splittings

Fitting, socle, SylowC53 ⋊ C4
derivedC52 ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC5#1044
commutator, hypocenterC52C4×C5
FrattiniC1G
Fitting, socleC53C4
prime radicalC5×C5⋊D5C2
cosocleC5⋊D5C2×C5
LCSG ▷ C52
UCSC1 ◁ C5

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 20

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2,3,4,5)(7,8,9,10)(12,13,14,15);(6,7,10,8,9)(11,12,13,15,14)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×D5⋊AGL(1,𝔽5), D5×#1044, #11758789, …
maximal normal subgroup ofC5×D5⋊AGL(1,𝔽5), D5×#1044, #11758789, …
maximal quotient ofC5×#6034, C2×C5×#1044, C3×C5×#1044, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 44);