Finite Group Databasebeta

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C22×AGL(1,𝔽5)×3−1+2 #72247330

order2160
automorphism?
homologyAbelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23

Aliases

direct productC22×AGL(1,𝔽5)×3−1+2

Splittings

FittingC22×C5×3−1+2 ⋊ C4
HallC5×3−1+2 ⋊ C22×C4
SylowC5 ⋊ C22×C4×3−1+2

Subgroups

subgroupG/H
centerC22×C3C32×AGL(1,𝔽5)
commutatorC3×C5C22×C32×C4
FrattiniC3C22×C32×AGL(1,𝔽5)
FittingC22×C5×3−1+2C4
socleC22×C3×C5C32×C4
prime radicalC22×C32×D5C2×C3
cosocleC3×D5C23×C32
hypercenterC22×3−1+2AGL(1,𝔽5)
hypocenterC5C22×C4×3−1+2
LCSG ▷ C3×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×3−1+2

432 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 224 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

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×AGL(1,𝔽5)×AGL(1,ℤ/9), C22×3−1+2×#1043, D4×AGL(1,𝔽5)×3−1+2, …
maximal normal subgroup ofC22×AGL(1,𝔽5)×AGL(1,ℤ/9), D4×AGL(1,𝔽5)×3−1+2, AGL(1,𝔽5)×#611⋊C3, …
maximal quotient ofC22×3−1+2×#1043, D4×AGL(1,𝔽5)×3−1+2, C22×AGL(1,𝔽5)×#1094, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.