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H(5,𝔽2)×ΓL(1,𝔽8)⋊AGL(1,𝔽7) #72246285

order28224
automorphism?
homologyAbelianization (H1) = C25×C32 · Schur multiplier (H2) = C29×C3

Aliases

direct productH(5,𝔽2)×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

Splittings

FittingC72×H(5,𝔽2) ⋊ C2×C32
HallH(5,𝔽2)×C7⋊D7 ⋊ C32
HallΓL(1,𝔽8)2 ⋊ C2×H(5,𝔽2)
SylowC72 ⋊ C2×C32×H(5,𝔽2)

Subgroups

subgroupG/H
center, FrattiniC2C24×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
commutator, socle, cosocleC2×C72C25×C32
FittingC72×H(5,𝔽2)C2×C32
hypercenterH(5,𝔽2)ΓL(1,𝔽8)⋊AGL(1,𝔽7)
hypocenterC72C2×C32×H(5,𝔽2)
LCSG ▷ C2×C72 ▷ C72
UCSC1 ◁ C2 ◁ H(5,𝔽2)

12572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

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

Relationships

maximal subgroup of#453×ΓL(1,𝔽8)⋊AGL(1,𝔽7), B(4,𝔽2)×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
maximal normal subgroup of#453×ΓL(1,𝔽8)⋊AGL(1,𝔽7), B(4,𝔽2)×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id.