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C5×ΓL(1,𝔽8)×D55 #26040503

order11550
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×ΓL(1,𝔽8)×D55
unambiguous semidirect productC5⋊(C5×ΓL(1,𝔽8)×D11)
unambiguous semidirect product(C5×C7×D55)⋊C3

Splittings

Fitting, socle, HallC52×C7×C11 ⋊ C2×C3
derivedC5×C7×C11 ⋊ C2×C3×C5
HallC52×C11×ΓL(1,𝔽8) ⋊ C2
HallC5×C7×D55 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ D11
HallC52×C11 ⋊ C2×ΓL(1,𝔽8)
HallC11×ΓL(1,𝔽8) ⋊ C5×D5
HallC52×C7 ⋊ C3×D11
HallC7×C11 ⋊ C3×C5×D5
SylowC52 ⋊ ΓL(1,𝔽8)×D11
SylowC11 ⋊ C5×D5×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C5×D55

Subgroups

subgroupG/H
center, hypercenterC5ΓL(1,𝔽8)×D55
commutator, cosocle, hypocenterC5×C7×C11C2×C3×C5
FrattiniC1G
Fitting, socleC52×C7×C11C2×C3
LCSG ▷ C5×C7×C11
UCSC1 ◁ C5

80 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

725 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×AGL(1,𝔽7)×D55, C5×D55⋊AGL(1,𝔽7), C5×D5×ΓL(1,𝔽8)×D11, …
maximal normal subgroup ofC5×AGL(1,𝔽7)×D55, C5×D55⋊AGL(1,𝔽7), C5×D5×ΓL(1,𝔽8)×D11, …
maximal quotient ofC5×ΓL(1,𝔽8)×Dic55, C2×C5×ΓL(1,𝔽8)×D55, C5×C7⋊C9×D55, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11550, 67);