Finite Group Databasebeta

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3−1+2×#87831 #72254658

order10800
automorphism?
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C22

Aliases

direct product3−1+2×#87831

Splittings

HallC52×3−1+2 ⋊ C22⋊C4
SylowC52 ⋊ C22⋊C4×3−1+2

Subgroups

subgroupG/H
center, FrattiniC2×C3C32×D5⋊AGL(1,𝔽5)
commutator, socleC2×C3×C52C2×C32×C4
FittingC2×C52×3−1+2C2×C4
prime radicalC2×C32×D52C2×C3
cosocleC2×C3×C5⋊D5C22×C32
hypercenterC2×3−1+2D5⋊AGL(1,𝔽5)
hypocenterC52C22⋊C4×3−1+2
LCSG ▷ C2×C3×C52 ▷ C52
UCSC1 ◁ C2×C3 ◁ C2×3−1+2

672 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 133 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) = 180

Conjugacy classes & character table

242 conjugacy classes — view the full character table.

Relationships

maximal subgroup of3−1+2×#11717375, AGL(1,ℤ/9)×#87831, #72254807
maximal normal subgroup of3−1+2×#11717375, AGL(1,ℤ/9)×#87831, #72254807

GAP

Past the SmallGroups library coverage — no SmallGroups id.