Finite Group Databasebeta

← Prev (#72247257) Next (#72247259) →

AGL(1,𝔽5)×C3⋊((C3⋊D3)⋊C3) #72247258

order3240
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C32

Aliases

direct productAGL(1,𝔽5)×C3⋊((C3⋊D3)⋊C3)

Splittings

Fitting, HallC3×C5×H(3,𝔽3) ⋊ C2×C4
derivedC33×C5 ⋊ C2×C3×C4
SylowC3×H(3,𝔽3) ⋊ C2×AGL(1,𝔽5)
SylowC5 ⋊ C4×C3⋊((C3⋊D3)⋊C3)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenterC33×C5C2×C3×C4
FrattiniC3C3×C3⋊D3×AGL(1,𝔽5)
FittingC3×C5×H(3,𝔽3)C2×C4
socleC32×C5C3×C4×D3
prime radicalD5×C3⋊((C3⋊D3)⋊C3)C2
cosocleC33×D5C22×C3
LCSG ▷ C33×C5
UCSC1

506 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 76 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

105 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×((C3⋊D3)⋊C3)⋊D3
maximal normal subgroup ofAGL(1,𝔽5)×((C3⋊D3)⋊C3)⋊D3
maximal quotient ofAGL(1,𝔽5)×#92300

GAP

Past the SmallGroups library coverage — no SmallGroups id.