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ΓL(1,𝔽8)×#4078 #72217976

order3360
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productΓL(1,𝔽8)×#4078

Splittings

FittingC2×C5×C7×D4 ⋊ C2×C3
HallC7×#4078 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ AGL(1,ℤ/8)
HallC5×C7 ⋊ C3×AGL(1,ℤ/8)
SylowC7 ⋊ C3×#4078
SylowC5 ⋊ ΓL(1,𝔽8)×AGL(1,ℤ/8)

Subgroups

subgroupG/H
centerC2C2×ΓL(1,𝔽8)×C22⋊D5
commutator, cosocleC4×C5×C7C23×C3
FrattiniC4C22×D5×ΓL(1,𝔽8)
FittingC2×C5×C7×D4C2×C3
socleC2×C5×C7C2×C3×D4
hypercenterC2×D4D5×ΓL(1,𝔽8)
hypocenterC5×C7C3×AGL(1,ℤ/8)
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×ΓL(1,𝔽8)×AGL(1,ℤ/8), AGL(1,𝔽7)×#4078, #72218497
maximal normal subgroup ofD5×ΓL(1,𝔽8)×AGL(1,ℤ/8), AGL(1,𝔽7)×#4078, #72218497

GAP

Past the SmallGroups library coverage — no SmallGroups id.