Finite Group Databasebeta

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Cayley diagram

C5×#1042 #92556

order500
isoclinismisoclinic class of #1042
automorphismAut(G) = C4×AGL(1,𝔽5)2 (order 1600) · Inn(G) = #1042 · Out(G) = C42
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×#1042

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC5#1042
commutator, hypocenterC52C4×C5
FrattiniC1G
Fitting, socleC53C4
prime radicalC52×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);(6,7,9,10,8)(11,12,15,13,14)

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×D5×AGL(1,𝔽5), D5×#1042, #1042⋊D5, …
maximal normal subgroup ofC5×D5×AGL(1,𝔽5), D5×#1042, #1042⋊D5, …
maximal quotient ofC5×#6032, C2×C5×#1042, C3×C5×#1042, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 42);