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C3×H(5,𝔽2)×AGL(1,𝔽7) #72244395

order4032
automorphism?
homologyAbelianization (H1) = C25×C32 · Schur multiplier (H2) = C29×C3

Aliases

direct productC3×H(5,𝔽2)×AGL(1,𝔽7)

Splittings

FittingC3×C7×H(5,𝔽2) ⋊ C2×C3
HallD7×H(5,𝔽2) ⋊ C32
HallC3×ΓL(1,𝔽8) ⋊ C2×H(5,𝔽2)
SylowC7 ⋊ C2×C32×H(5,𝔽2)

Subgroups

subgroupG/H
centerC2×C3C24×AGL(1,𝔽7)
commutator, cosocleC2×C7C25×C32
FrattiniC2C24×C3×AGL(1,𝔽7)
FittingC3×C7×H(5,𝔽2)C2×C3
socleC2×C3×C7C25×C3
hypercenterC3×H(5,𝔽2)AGL(1,𝔽7)
hypocenterC7C2×C32×H(5,𝔽2)
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C3×H(5,𝔽2)

5388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

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

Relationships

maximal subgroup ofD3×H(5,𝔽2)×AGL(1,𝔽7), C3×AGL(1,𝔽7)×#997, C3×AGL(1,𝔽7)×#453, …
maximal normal subgroup ofD3×H(5,𝔽2)×AGL(1,𝔽7), C3×AGL(1,𝔽7)×#997, C3×AGL(1,𝔽7)×#453, …
maximal quotient ofH(5,𝔽2)×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id.