Finite Group Databasebeta

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AGL(1,𝔽5)×#92300 #72247262

order9720
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C32

Aliases

direct productAGL(1,𝔽5)×#92300

Splittings

Fitting, HallC5×C32⋊H(3,𝔽3) ⋊ C2×C4
derivedC3×C5×H(3,𝔽3) ⋊ C2×C3×C4
SylowC32⋊H(3,𝔽3) ⋊ C2×AGL(1,𝔽5)
SylowC5 ⋊ C4×#92300

Subgroups

subgroupG/H
center, hypercenterC3AGL(1,𝔽5)×C3⋊((C3⋊D3)⋊C3)
commutator, hypocenterC3×C5×H(3,𝔽3)C2×C3×C4
FrattiniC32C3×C3⋊D3×AGL(1,𝔽5)
FittingC5×C32⋊H(3,𝔽3)C2×C4
socleC32×C5C3×C4×C3⋊D3
prime radicalD5×#92300C2
cosocleC3×D5×H(3,𝔽3)C22×C3
LCSG ▷ C3×C5×H(3,𝔽3)
UCSC1 ◁ C3

1412 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 1412 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 90

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#13091120, AGL(1,𝔽5)×#11757730
maximal normal subgroup ofAGL(1,𝔽5)×#11757730

GAP

Past the SmallGroups library coverage — no SmallGroups id.