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Cayley diagram

C5×#157 #4378

order180
isoclinismisoclinic class of #157
automorphismAut(G) = C4×AΓL(1,𝔽9) (order 576) · Inn(G) = #157 · Out(G) = C22×C4
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×#157

Splittings

Fitting, socle, HallC32×C5 ⋊ C4
derived, SylowC32 ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC5#157
commutator, hypocenterC32C4×C5
FrattiniC1G
Fitting, socleC32×C5C4
prime radicalC5×C3⋊D3C2
cosocleC3⋊D3C2×C5
LCSG ▷ C32
UCSC1 ◁ C5

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 30

A minimal faithful permutation representation — 2 generators on 11 points:

(1,2)(3,5,6,4)(7,8,9,10,11);(1,3,6)(2,5,4)

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

derived subgroup of#10624311, PSU(3,𝔽2)⋊D5, AGL(1,𝔽9)⋊D5, …
maximal subgroup ofC5×AGL(1,𝔽9), #67194, D5×#157, …
maximal normal subgroup ofC5×AGL(1,𝔽9), #67194, D5×#157, …
maximal quotient ofC5×#622, C2×C5×#157, C5×#1088, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 23);