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C2×AGL(1,𝔽5)×H(3,𝔽3) #11761970

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

Aliases

direct productC2×AGL(1,𝔽5)×H(3,𝔽3)
unambiguous semidirect productD5⋊(C4×H(3,𝔽3))
unambiguous semidirect product(C2×C32×AGL(1,𝔽5))⋊C3

Splittings

FittingC2×C5×H(3,𝔽3) ⋊ C4
HallC5×H(3,𝔽3) ⋊ C2×C4
SylowC5 ⋊ C2×C4×H(3,𝔽3)

Subgroups

subgroupG/H
centerC2×C3C32×AGL(1,𝔽5)
commutatorC3×C5C2×C32×C4
FrattiniC3C2×C32×AGL(1,𝔽5)
FittingC2×C5×H(3,𝔽3)C4
socleC2×C3×C5C32×C4
prime radicalC2×D5×H(3,𝔽3)C2
cosocleC3×D5C22×C32
hypercenterC2×H(3,𝔽3)AGL(1,𝔽5)
hypocenterC5C2×C4×H(3,𝔽3)
LCSG ▷ C3×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C2×H(3,𝔽3)

176 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)×(C3⋊D3)⋊C3, C2×C3×AGL(1,𝔽5)×H(3,𝔽3), C2×AGL(1,𝔽5)×#272, …
maximal normal subgroup ofC2×AGL(1,𝔽5)×(C3⋊D3)⋊C3, C2×C3×AGL(1,𝔽5)×H(3,𝔽3), C2×AGL(1,𝔽5)×#272
maximal quotient ofC2×C3×AGL(1,𝔽5)×H(3,𝔽3), D3×AGL(1,𝔽5)×H(3,𝔽3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1080, 316);