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C11⋊C5×ΓL(1,𝔽8)⋊D7 #30633505

order16170
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC11⋊C5×ΓL(1,𝔽8)⋊D7
unambiguous semidirect productΓL(1,𝔽8)⋊(D7×C11⋊C5)
unambiguous semidirect product(C11×ΓL(1,𝔽8)⋊D7)⋊C5

Splittings

Fitting, socle, derived, HallC72×C11 ⋊ C2×C3×C5
HallC7×ΓL(1,𝔽8)×C11⋊C5 ⋊ C2
HallC11⋊C5×C7⋊D7 ⋊ C3
HallC11×ΓL(1,𝔽8)⋊D7 ⋊ C5
HallC72×C11⋊C5 ⋊ C2×C3
HallC7×C11×ΓL(1,𝔽8) ⋊ C2×C5
HallC11×C7⋊D7 ⋊ C3×C5
HallC7×ΓL(1,𝔽8) ⋊ C2×C11⋊C5
HallC7⋊D7 ⋊ C3×C11⋊C5
SylowC72 ⋊ C2×C3×C11⋊C5
SylowC11 ⋊ C5×ΓL(1,𝔽8)⋊D7

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72×C11C2×C3×C5
LCSG ▷ C72×C11
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 231

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽11)×ΓL(1,𝔽8)⋊D7, (ΓL(1,𝔽8)⋊D7)⋊AGL(1,𝔽11), D7×AGL(1,𝔽7)×C11⋊C5, …
maximal normal subgroup ofAGL(1,𝔽11)×ΓL(1,𝔽8)⋊D7, (ΓL(1,𝔽8)⋊D7)⋊AGL(1,𝔽11), D7×AGL(1,𝔽7)×C11⋊C5, …
maximal quotient ofC11⋊C5×ΓL(1,𝔽8)⋊Dic7, C2×C11⋊C5×ΓL(1,𝔽8)⋊D7

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 50);