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C2×AGL(1,𝔽5)×3−1+2 #11761971

order1080
automorphism?
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×AGL(1,𝔽5)×3−1+2
unambiguous semidirect productD5⋊(C4×3−1+2)
unambiguous semidirect product(C2×C9×AGL(1,𝔽5))⋊C3

Splittings

FittingC2×C5×3−1+2 ⋊ C4
HallC5×3−1+2 ⋊ C2×C4
SylowC5 ⋊ C2×C4×3−1+2

Subgroups

subgroupG/H
centerC2×C3C32×AGL(1,𝔽5)
commutatorC3×C5C2×C32×C4
FrattiniC3C2×C32×AGL(1,𝔽5)
FittingC2×C5×3−1+2C4
socleC2×C3×C5C32×C4
prime radicalC2×C32×D5C2×C3
cosocleC3×D5C22×C32
hypercenterC2×3−1+2AGL(1,𝔽5)
hypocenterC5C2×C4×3−1+2
LCSG ▷ C3×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×3−1+2

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 90

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×AGL(1,𝔽5)×AGL(1,ℤ/9), 3−1+2×D5⋊AGL(1,𝔽5), D3×AGL(1,𝔽5)×3−1+2, …
maximal normal subgroup ofC2×AGL(1,𝔽5)×AGL(1,ℤ/9), C22×AGL(1,𝔽5)×3−1+2, C4×AGL(1,𝔽5)×3−1+2, …
maximal quotient of3−1+2×D5⋊AGL(1,𝔽5), D3×AGL(1,𝔽5)×3−1+2, C2×3−1+2×#1043, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1080, 317);