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ΓL(1,𝔽8)×#3994 #72217751

order3360
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productΓL(1,𝔽8)×#3994

Splittings

FittingC5×C7×C8 ⋊ C3×C4
HallC7×#3994 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ #107
HallC5×C7 ⋊ C3×#107
SylowC7 ⋊ C3×#3994
SylowC5 ⋊ ΓL(1,𝔽8)×#107

Subgroups

subgroupG/H
centerC2ΓL(1,𝔽8)×C4⋊AGL(1,𝔽5)
commutatorC4×C5×C7C2×C3×C4
FrattiniC4C2×AGL(1,𝔽5)×ΓL(1,𝔽8)
FittingC5×C7×C8C3×C4
socleC2×C5×C7C3×C4⋊C4
prime radicalC2×D5×ΓL(1,𝔽8)D4
cosocleC4×C7×D5C22×C3
hypercenterC8AGL(1,𝔽5)×ΓL(1,𝔽8)
hypocenterC5×C7C3×#107
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C4 ◁ C8

144 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 54 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

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD8×AGL(1,𝔽5)×ΓL(1,𝔽8), #72218038, AGL(1,𝔽7)×#3994, …
maximal normal subgroup ofD8×AGL(1,𝔽5)×ΓL(1,𝔽8), #72218038, AGL(1,𝔽7)×#3994, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.