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C2×C5×C7×C11×ΓL(1,𝔽8) #30633569

order16170
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C3×C5×C7×C11 · Schur multiplier (H2) = C1

Aliases

direct productC2×C5×C7×C11×ΓL(1,𝔽8)
unambiguous semidirect product(C2×C7)⋊(C3×C5×C7×C11)
unambiguous semidirect product(C2×C5×C7)⋊(C3×C7×C11)
unambiguous semidirect product(C2×C5×C7×C11)⋊(C3×C7)

Splittings

Fitting, socle, HallC2×C5×C72×C11 ⋊ C3
derivedC7 ⋊ C2×C3×C5×C7×C11
HallC5×C72×C11 ⋊ C2×C3
HallC2×C72×C11 ⋊ C3×C5
HallC72×C11 ⋊ C2×C3×C5
HallC2×C5×C72 ⋊ C3×C11
HallC5×C72 ⋊ C2×C3×C11
HallC2×C72 ⋊ C3×C5×C11
SylowC72 ⋊ C2×C3×C5×C11

Subgroups

subgroupG/H
center, hypercenterC2×C5×C7×C11ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C3×C5×C7×C11
FrattiniC1G
Fitting, socleC2×C5×C72×C11C3
LCSG ▷ C7
UCSC1 ◁ C2×C5×C7×C11

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2310

Conjugacy classes & character table

3850 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC5×C7×C11×ΓL(1,𝔽8)⋊C4, C5×C7×C11⋊(ΓL(1,𝔽8)⋊C4), C5×C7×ΓL(1,𝔽8)×Dic11, …
maximal normal subgroup ofC5×C7×C11×ΓL(1,𝔽8)⋊C4, C5×C7×C11⋊(ΓL(1,𝔽8)⋊C4), C5×C7×ΓL(1,𝔽8)×Dic11, …
maximal quotient ofC4×C5×C7×C11×ΓL(1,𝔽8), C22×C5×C7×C11×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 114);