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AGL(1,𝔽61) #16814567

order3660
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

affine general linearAGL(1,𝔽61)
unambiguous semidirect product(#6923⋊C3)⋊C5
unambiguous semidirect product(#6923⋊C5)⋊C3

Splittings

Fitting, socle, derived, SylowC61 ⋊ C3×C4×C5
Hall#6923⋊C5 ⋊ C3
Hall(C61⋊C3)⋊C5 ⋊ C4
Hall#6923⋊C3 ⋊ C5
HallC61⋊C5 ⋊ C3×C4
Hall#6923 ⋊ C3×C5
HallC61⋊C3 ⋊ C4×C5

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC61C3×C4×C5
prime radical(D61⋊C3)⋊C5C2
cosocleD61C2×C3×C5
LCSG ▷ C61
UCSC1

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 61
transitive degreeμt(G) = 61

Polycyclic presentation:

2,3,5,2,61|a^2=d,e^a=e^50,e^b=e^13,e^c=e^9,e^d=e^60

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽61), C3⋊AGL(1,𝔽61), C5×AGL(1,𝔽61), …
maximal normal subgroup ofC3×AGL(1,𝔽61), C5×AGL(1,𝔽61), C7×AGL(1,𝔽61), …
maximal quotient of(#6923⋊C5)⋊C9, C3×AGL(1,𝔽61), C3⋊AGL(1,𝔽61), …
automorphism group ofD61, C61⋊C3, #6923
holomorph ofC61

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3660, 20);