Finite Group Databasebeta

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ΓL(1,𝔽8)×#65793 #72218324

order6720
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productΓL(1,𝔽8)×#65793

Splittings

FittingC5×C7×#41 ⋊ C3×C4
HallC7×#65793 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ #417
HallC5×C7 ⋊ C3×#417
SylowC7 ⋊ C3×#65793
SylowC5 ⋊ ΓL(1,𝔽8)×#417

Subgroups

subgroupG/H
centerC2C2×ΓL(1,𝔽8)×C22⋊AGL(1,𝔽5)
commutatorC4×C5×C7C22×C3×C4
FrattiniC4C22×AGL(1,𝔽5)×ΓL(1,𝔽8)
FittingC5×C7×#41C3×C4
socleC2×C5×C7C2×C3×C22⋊C4
prime radicalD5×#41×ΓL(1,𝔽8)C2
cosocleC4×C7×D5C23×C3
hypercenter#41AGL(1,𝔽5)×ΓL(1,𝔽8)
hypocenterC5×C7C3×#417
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

648 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×ΓL(1,𝔽8)×AGL(1,ℤ/8), #72218611, AGL(1,𝔽7)×#65793
maximal normal subgroup ofAGL(1,𝔽5)×ΓL(1,𝔽8)×AGL(1,ℤ/8), #72218611, AGL(1,𝔽7)×#65793

GAP

Past the SmallGroups library coverage — no SmallGroups id.