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Cayley diagram

C3×D3×AGL(1,𝔽5) #67196

order360
isoclinismisoclinic class of D3×AGL(1,𝔽5)
automorphismAut(G) = C2×D3×AGL(1,𝔽5) (order 240) · Inn(G) = D3×AGL(1,𝔽5) · Out(G) = C2
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×D3×AGL(1,𝔽5)
unambiguous semidirect product(C3×AGL(1,𝔽5))⋊(C2×C3)
unambiguous semidirect productΓL(1,𝔽16)⋊(C2×C3)

Splittings

Fitting, socle, HallC32×C5 ⋊ C2×C4
derivedC3×C5 ⋊ C2×C3×C4
SylowC32 ⋊ C2×AGL(1,𝔽5)
SylowC5 ⋊ C3×C4×D3

Subgroups

subgroupG/H
center, hypercenterC3D3×AGL(1,𝔽5)
commutator, hypocenterC3×C5C2×C3×C4
FrattiniC1G
Fitting, socleC32×C5C2×C4
prime radicalC3×D3×D5C2
cosocleC3×D5C22×C3
LCSG ▷ C3×C5
UCSC1 ◁ C3

70 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)(4,5,6)(8,11,10,9);(5,6)(7,8,9,11,10)

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD32×AGL(1,𝔽5), C2×C3×D3×AGL(1,𝔽5), C3×D9×AGL(1,𝔽5), …
maximal normal subgroup ofD32×AGL(1,𝔽5), C2×C3×D3×AGL(1,𝔽5), D3×C9×AGL(1,𝔽5), …
maximal quotient ofC3×Dic3×AGL(1,𝔽5), C3×(C2×D3)⋊AGL(1,𝔽5), C3×Dic3⋊AGL(1,𝔽5), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 126);