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C8×AGL(1,𝔽5)×ΓL(1,𝔽8) #72217366

order3360
automorphism?
homologyAbelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C4

Aliases

direct productC8×AGL(1,𝔽5)×ΓL(1,𝔽8)

Splittings

FittingC5×C7×C8 ⋊ C3×C4
derived, HallC5×C7 ⋊ C3×C4×C8
HallC7×C8×AGL(1,𝔽5) ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C4×C8
SylowC7 ⋊ C3×C8×AGL(1,𝔽5)
SylowC5 ⋊ C4×C8×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC8AGL(1,𝔽5)×ΓL(1,𝔽8)
commutator, hypocenterC5×C7C3×C4×C8
FrattiniC4C2×AGL(1,𝔽5)×ΓL(1,𝔽8)
FittingC5×C7×C8C3×C4
socleC2×C5×C7C3×C42
prime radicalC2×D5×ΓL(1,𝔽8)C2×C4
cosocleC4×C7×D5C22×C3
LCSG ▷ C5×C7
UCSC1 ◁ C8

176 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 78 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

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×ΓL(1,𝔽8)×AGL(1,𝔽9), C8×AGL(1,𝔽5)×AGL(1,𝔽7), M16×AGL(1,𝔽5)×ΓL(1,𝔽8), …
maximal normal subgroup ofC8×AGL(1,𝔽5)×AGL(1,𝔽7), M16×AGL(1,𝔽5)×ΓL(1,𝔽8), AGL(1,𝔽5)×#1131⋊C3, …
maximal quotient ofAGL(1,𝔽5)×ΓL(1,𝔽8)×AGL(1,𝔽9)

GAP

Past the SmallGroups library coverage — no SmallGroups id.