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Cayley diagram

C5×GO+(4,𝔽2) #67202

order360
isoclinismisoclinic class of GO+(4,𝔽2)
automorphismAut(G) = C4×AΓL(1,𝔽9) (order 576) · Inn(G) = GO+(4,𝔽2) · Out(G) = C2×C4
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×GO+(4,𝔽2)
unambiguous semidirect product#157⋊(C2×C5)
unambiguous semidirect productD32⋊(C2×C5)

Splittings

Fitting, socle, HallC32×C5 ⋊ D4
SylowC32 ⋊ C5×D4

Subgroups

subgroupG/H
center, hypercenterC5GO+(4,𝔽2)
commutator, cosocleC3⋊D3C22×C5
FrattiniC1G
Fitting, socleC32×C5D4
hypocenterC32C5×D4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C5

52 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:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×AΓL(1,𝔽9), #10624311, D5×GO+(4,𝔽2), …
maximal normal subgroup ofC5×AΓL(1,𝔽9), #10624311, D5×GO+(4,𝔽2), …
maximal quotient ofC5×D32⋊C4, C5×#3773, C5×#3774, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 132);