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ΓL(1,𝔽8)×D255 #24924225

order10710
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D255
unambiguous semidirect productC3⋊(ΓL(1,𝔽8)×D85)
unambiguous semidirect productC5⋊(ΓL(1,𝔽8)×D51)
unambiguous semidirect productC17⋊(ΓL(1,𝔽8)×D15)
unambiguous semidirect product(C7×D255)⋊C3

Splittings

Fitting, socle, derivedC3×C5×C7×C17 ⋊ C2×C3
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C2
HallC3×C17×ΓL(1,𝔽8) ⋊ D5
HallC5×C7×C17 ⋊ C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ D17
HallC7×C17 ⋊ C3×D15
HallC5×C17 ⋊ D3×ΓL(1,𝔽8)
HallC3×ΓL(1,𝔽8) ⋊ D85
HallC5×C7 ⋊ C3×D51
SylowC17 ⋊ ΓL(1,𝔽8)×D15
SylowC7 ⋊ C3×D255
SylowC5 ⋊ ΓL(1,𝔽8)×D51

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C7×C17C2×C3
LCSG ▷ C3×C5×C7×C17
UCSC1

72 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

645 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×D255, (C17⋊AGL(1,𝔽7))⋊D15, ΓL(1,𝔽8)×D15×D17, …
maximal normal subgroup ofAGL(1,𝔽7)×D255, (C17⋊AGL(1,𝔽7))⋊D15, ΓL(1,𝔽8)×D15×D17, …
maximal quotient ofΓL(1,𝔽8)×Dic255, C2×ΓL(1,𝔽8)×D255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10710, 40);