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C22×D5×3−1+2 #11762001

order1080
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C23

Aliases

direct productC22×D5×3−1+2
unambiguous semidirect productC5⋊(C23×3−1+2)
unambiguous semidirect product(C2×C5)⋊(C22×3−1+2)

Splittings

FittingC22×C5×3−1+2 ⋊ C2
HallC5×3−1+2 ⋊ C23
SylowC5 ⋊ C23×3−1+2

Subgroups

subgroupG/H
centerC22×C3C32×D5
commutator, cosocleC3×C5C23×C32
FrattiniC3C22×C32×D5
FittingC22×C5×3−1+2C2
socleC22×C3×C5C2×C32
prime radicalC22×C32×D5C3
hypercenterC22×3−1+2D5
hypocenterC5C23×3−1+2
LCSG ▷ C3×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×3−1+2

256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 180

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×AGL(1,𝔽5)×3−1+2, C22×D5×AGL(1,ℤ/9), C22×3−1+2⋊AGL(1,𝔽5), …
maximal normal subgroup ofC22×AGL(1,𝔽5)×3−1+2, C22×D5×AGL(1,ℤ/9), C22×3−1+2⋊AGL(1,𝔽5), …
maximal quotient ofC22×3−1+2×C5⋊D5, D4×D5×3−1+2, C22×D5×#1094, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1080, 347);