Finite Group Databasebeta

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C3×#120×AGL(1,𝔽7) #72244143

order4032
automorphism?
homologyAbelianization (H1) = C24×C32 · Schur multiplier (H2) = C27×C3

Aliases

direct productC3×#120×AGL(1,𝔽7)

Splittings

FittingC3×C7×#120 ⋊ C2×C3
HallD7×#120 ⋊ C32
HallC3×ΓL(1,𝔽8) ⋊ C2×#120
SylowC7 ⋊ C2×C32×#120

Subgroups

subgroupG/H
centerC22×C3C23×AGL(1,𝔽7)
commutator, cosocleC22×C7C24×C32
FrattiniC22C23×C3×AGL(1,𝔽7)
FittingC3×C7×#120C2×C3
socleC22×C3×C7C24×C3
hypercenterC3×#120AGL(1,𝔽7)
hypocenterC7C2×C32×#120
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C3×#120

3972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

294 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD3×#120×AGL(1,𝔽7), #120×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C3×AGL(1,𝔽7)×B(4,𝔽2), …
maximal normal subgroup ofD3×#120×AGL(1,𝔽7), C3×AGL(1,𝔽7)×B(4,𝔽2), AGL(1,𝔽7)×#5032, …
maximal quotient ofC3×AGL(1,𝔽7)×#453, #120×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C3×AGL(1,𝔽7)×B(4,𝔽2), …

GAP

Past the SmallGroups library coverage — no SmallGroups id.