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ΓL(1,𝔽8)×D187 #21463282

order7854
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D187
unambiguous semidirect productC11⋊(ΓL(1,𝔽8)×D17)
unambiguous semidirect productC17⋊(ΓL(1,𝔽8)×D11)
unambiguous semidirect product(C7×D187)⋊C3

Splittings

Fitting, socle, derived, HallC7×C11×C17 ⋊ C2×C3
HallC11×C17×ΓL(1,𝔽8) ⋊ C2
HallC7×D187 ⋊ C3
HallC17×ΓL(1,𝔽8) ⋊ D11
HallC11×ΓL(1,𝔽8) ⋊ D17
HallC11×C17 ⋊ C2×ΓL(1,𝔽8)
HallC7×C17 ⋊ C3×D11
HallC7×C11 ⋊ C3×D17
SylowC17 ⋊ ΓL(1,𝔽8)×D11
SylowC11 ⋊ ΓL(1,𝔽8)×D17
SylowC7 ⋊ C3×D187

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC7×C11×C17C2×C3
LCSG ▷ C7×C11×C17
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

475 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×D187, D187⋊AGL(1,𝔽7), ΓL(1,𝔽8)×D11×D17, …
maximal normal subgroup ofAGL(1,𝔽7)×D187, D187⋊AGL(1,𝔽7), ΓL(1,𝔽8)×D11×D17, …
maximal quotient ofΓL(1,𝔽8)×Dic187, C2×ΓL(1,𝔽8)×D187, C7⋊C9×D187, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7854, 20);