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#120×C3⋊AGL(1,𝔽7) #72244174

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

Aliases

direct product#120×C3⋊AGL(1,𝔽7)

Splittings

FittingC3×C7×#120 ⋊ C2×C3
HallC3×ΓL(1,𝔽8) ⋊ C2×#120
SylowC7 ⋊ C3×D3×#120

Subgroups

subgroupG/H
center, FrattiniC22C23×C3⋊AGL(1,𝔽7)
commutator, socle, cosocleC22×C3×C7C24×C3
FittingC3×C7×#120C2×C3
hypercenter#120C3⋊AGL(1,𝔽7)
hypocenterC3×C7C2×C3×#120
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #120

2778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#120×AGL(1,𝔽7), B(4,𝔽2)×C3⋊AGL(1,𝔽7), #72269452, …
maximal normal subgroup ofD3×#120×AGL(1,𝔽7), B(4,𝔽2)×C3⋊AGL(1,𝔽7), #72269452
maximal quotient of#453×C3⋊AGL(1,𝔽7), B(4,𝔽2)×C3⋊AGL(1,𝔽7), #120×C3⋊(C3⋊AGL(1,𝔽7))

GAP

Past the SmallGroups library coverage — no SmallGroups id.