Finite Group Databasebeta

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M16×ΓL(1,𝔽8)⋊AGL(1,𝔽7) #72245299

order14112
automorphism?
homologyAbelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C22×C3

Aliases

direct productM16×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

Splittings

FittingC72×M16 ⋊ C2×C32
HallM16×C7⋊D7 ⋊ C32
HallΓL(1,𝔽8)2 ⋊ C2×M16
SylowC72 ⋊ C2×C32×M16

Subgroups

subgroupG/H
center, FrattiniC4C22×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
commutator, socleC2×C72C22×C32×C4
FittingC72×M16C2×C32
prime radicalC22×ΓL(1,𝔽8)⋊AGL(1,𝔽7)C4
cosocleC4×C72C23×C32
hypercenterM16ΓL(1,𝔽8)⋊AGL(1,𝔽7)
hypocenterC72C2×C32×M16
LCSG ▷ C2×C72 ▷ C72
UCSC1 ◁ C4 ◁ M16

952 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), #100×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), #100×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #104×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id.