Finite Group Databasebeta

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AGL(1,𝔽5)×#4178 #72268705

order3240
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productAGL(1,𝔽5)×#4178

Splittings

Fitting, HallC5×#732 ⋊ C2×C4
Sylow#732 ⋊ C2×AGL(1,𝔽5)
SylowC5 ⋊ C4×#4178

Subgroups

subgroupG/H
center, hypercenterC3AGL(1,𝔽5)×(C3⋊D3)⋊C3
commutator, hypocenterC5×H(3,𝔽3)C2×C3×C4
FrattiniC32C3×D3×AGL(1,𝔽5)
FittingC5×#732C2×C4
socleC3×C5C4×(C3⋊D3)⋊C3
prime radicalD5×#4178C2
cosocleD5×H(3,𝔽3)C22×C3
LCSG ▷ C5×H(3,𝔽3)
UCSC1 ◁ C3

178 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 135

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#4178⋊C2, AGL(1,𝔽5)×#92285
maximal normal subgroup ofAGL(1,𝔽5)×#4178⋊C2, AGL(1,𝔽5)×#92285

GAP

Past the SmallGroups library coverage — no SmallGroups id.