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Cayley diagram

C5×#261 #63117

order260
isoclinismisoclinic class of #261
automorphismAut(G) = C4×AGL(1,𝔽13) (order 624) · Inn(G) = #261 · Out(G) = C3×C4
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×#261

Splittings

Fitting, socle, HallC5×C13 ⋊ C4
derived, SylowC13 ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC5#261
commutator, hypocenterC13C4×C5
FrattiniC1G
Fitting, socleC5×C13C4
prime radicalC5×D13C2
cosocleD13C2×C5
LCSG ▷ C13
UCSC1 ◁ C5

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 65

A minimal faithful permutation representation — 2 generators on 18 points:

(1,2,3,4,5)(7,8,9,11)(10,13,14,18)(12,16,17,15);(6,7,10,13,17,11,15,16,8,12,18,14,9)

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×#261, C2×C5×#261, C5×AGL(1,𝔽13), …
maximal normal subgroup ofD5×#261, C2×C5×#261, C5×AGL(1,𝔽13), …
maximal quotient ofC5×#1057, C2×C5×#261, C3×C5×#261, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(260, 5);