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C2×ΓL(1,𝔽8)×C5⋊D5 #14523619

order2100
isoclinismisoclinic class of ΓL(1,𝔽8)×C5⋊D5
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C5

Aliases

direct productC2×ΓL(1,𝔽8)×C5⋊D5
unambiguous semidirect product(C2×C5)⋊(D5×ΓL(1,𝔽8))

Splittings

Fitting, socleC2×C52×C7 ⋊ C2×C3
derived, HallC52×C7 ⋊ C22×C3
HallC2×C7×C5⋊D5 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ C22
SylowC52 ⋊ C22×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C3×C5⋊D5

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×C5⋊D5
commutator, cosocle, hypocenterC52×C7C22×C3
FrattiniC1G
Fitting, socleC2×C52×C7C2×C3
LCSG ▷ C52×C7
UCSC1 ◁ C2

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 350

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C5⋊D5×C49⋊C3, C5⋊D5×#64423, C5⋊D5×#64422, …
maximal normal subgroup ofC2×C7×ΓL(1,𝔽8)×C5⋊D5, C2×C11×ΓL(1,𝔽8)×C5⋊D5, C2×C13×ΓL(1,𝔽8)×C5⋊D5, …
maximal quotient ofC2×C5⋊D5×C49⋊C3, C5⋊D5×#64423, C5⋊D5×#64422, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 99);