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#104×ΓL(1,𝔽8)⋊AGL(1,𝔽7) #72246392

order28224
automorphism?
homologyAbelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct product#104×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

Splittings

FittingC72×#104 ⋊ C2×C32
Hall#104×C7⋊D7 ⋊ C32
HallΓL(1,𝔽8)2 ⋊ C2×#104
SylowC72 ⋊ C2×C32×#104

Subgroups

subgroupG/H
centerC4D4×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
commutatorC4×C72C22×C32×C4
FrattiniC2×C4C22×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
FittingC72×#104C2×C32
socleC2×C72C2×C32×C22⋊C4
prime radical#41×ΓL(1,𝔽8)⋊AGL(1,𝔽7)C2
cosocleC2×C4×C72C23×C32
hypercenter#104ΓL(1,𝔽8)⋊AGL(1,𝔽7)
hypocenterC72C2×C32×#104
LCSG ▷ C4×C72 ▷ C2×C72 ▷ C72
UCSC1 ◁ C4 ◁ C2×C4 ◁ #104

2380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

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

Relationships

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

GAP

Past the SmallGroups library coverage — no SmallGroups id.