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C3×C52×C11×ΓL(1,𝔽8) #31773161

order17325
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C32×C52×C11 · Schur multiplier (H2) = C3×C5

Aliases

direct productC3×C52×C11×ΓL(1,𝔽8)
unambiguous semidirect productC7⋊(C32×C52×C11)
unambiguous semidirect product(C3×C7)⋊(C3×C52×C11)
unambiguous semidirect product(C5×C7)⋊(C32×C5×C11)
unambiguous semidirect product(C3×C5×C7)⋊(C3×C5×C11)

Splittings

Fitting, socleC3×C52×C7×C11 ⋊ C3
derived, SylowC7 ⋊ C32×C52×C11
HallC52×C7×C11 ⋊ C32
HallC52×C7 ⋊ C32×C11
HallC7×C11 ⋊ C32×C52

Subgroups

subgroupG/H
center, hypercenterC3×C52×C11ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C32×C52×C11
FrattiniC1G
Fitting, socleC3×C52×C7×C11C3
LCSG ▷ C7
UCSC1 ◁ C3×C52×C11

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5775

Conjugacy classes & character table

4125 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×C52×C11×AGL(1,𝔽7), C3×C52×C11⋊AGL(1,𝔽7), C3×C52×ΓL(1,𝔽8)×D11, …
maximal normal subgroup ofC3×C52×C11×AGL(1,𝔽7), C3×C52×C11⋊AGL(1,𝔽7), C3×C52×ΓL(1,𝔽8)×D11, …
maximal quotient ofC2×C3×C52×C11×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17325, 21);