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#41×ΓL(1,𝔽8)⋊AGL(1,𝔽7) #72245342

order14112
automorphism?
homologyAbelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3

Aliases

direct product#41×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

Splittings

FittingC72×#41 ⋊ C2×C32
Hall#41×C7⋊D7 ⋊ C32
HallΓL(1,𝔽8)2 ⋊ C2×#41
SylowC72 ⋊ C2×C32×#41

Subgroups

subgroupG/H
centerC4C22×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
commutator, socle, cosocleC2×C72C24×C32
FrattiniC2C23×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
FittingC72×#41C2×C32
hypercenter#41ΓL(1,𝔽8)⋊AGL(1,𝔽7)
hypocenterC72C2×C32×#41
LCSG ▷ C2×C72 ▷ C72
UCSC1 ◁ C4 ◁ #41

2296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

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

Relationships

maximal subgroup ofAGL(1,ℤ/8)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), H(5,𝔽2)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #104×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
maximal normal subgroup ofAGL(1,ℤ/8)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), H(5,𝔽2)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #104×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id.