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C2×ΓL(1,𝔽8)×D15 #11929528

order1260
isoclinismisoclinic class of ΓL(1,𝔽8)×D15
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×ΓL(1,𝔽8)×D15
unambiguous semidirect product(C2×C3)⋊(D5×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C5)⋊(D3×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×D15)⋊C3

Splittings

Fitting, socleC2×C3×C5×C7 ⋊ C2×C3
derivedC3×C5×C7 ⋊ C22×C3
HallC3×C5×ΓL(1,𝔽8) ⋊ C22
HallC3×ΓL(1,𝔽8) ⋊ C2×D5
HallC5×C7 ⋊ C2×C3×D3
SylowC7 ⋊ C2×C3×D15
SylowC5 ⋊ C2×D3×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×D15
commutator, cosocle, hypocenterC3×C5×C7C22×C3
FrattiniC1G
Fitting, socleC2×C3×C5×C7C2×C3
LCSG ▷ C3×C5×C7
UCSC1 ◁ C2

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 210

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×D11×D15, C2×C11×ΓL(1,𝔽8)×D15, C2×ΓL(1,𝔽8)×D165, …
maximal normal subgroup ofC2×C11×ΓL(1,𝔽8)×D15, C2×C13×ΓL(1,𝔽8)×D15, C2×C17×ΓL(1,𝔽8)×D15, …
maximal quotient ofΓL(1,𝔽8)×D11×D15, C2×C11×ΓL(1,𝔽8)×D15, C2×ΓL(1,𝔽8)×D165, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 80);