Finite Group Databasebeta

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GL(2,𝔽3)×ΓL(1,𝔽8)⋊AGL(1,𝔽5) #72218828

order20160
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productGL(2,𝔽3)×ΓL(1,𝔽8)⋊AGL(1,𝔽5)

Splittings

FittingC5×C7×Q8 ⋊ C3×C4×D3
derivedC5×C7×SL(2,𝔽3) ⋊ C2×C3×C4
HallC5×C7 ⋊ C3×C4×GL(2,𝔽3)
SylowC7 ⋊ C3×AGL(1,𝔽5)×GL(2,𝔽3)
SylowC5 ⋊ GL(2,𝔽3)×ΓL(1,𝔽8)⋊C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2S4×ΓL(1,𝔽8)⋊AGL(1,𝔽5)
commutator, hypocenterC5×C7×SL(2,𝔽3)C2×C3×C4
FittingC5×C7×Q8C3×C4×D3
socleC2×C5×C7C3×C4×S4
prime radicalD5×ΓL(1,𝔽8)×GL(2,𝔽3)C2
cosocleC7×D5×SL(2,𝔽3)C22×C3
LCSG ▷ C5×C7×SL(2,𝔽3)
UCSC1 ◁ C2

748 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 748 conjugacy classes of subgroups — too many to draw. Only its 65 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

152 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id.