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C7×ΓL(1,𝔽8)×C25 #16814719

order3675
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C7×C25 · Schur multiplier (H2) = C1

Aliases

direct productC7×ΓL(1,𝔽8)×C25
unambiguous semidirect productC7⋊(C3×C7×C25)
unambiguous semidirect product(C7×C25)⋊(C3×C7)

Splittings

Fitting, HallC72×C25 ⋊ C3
derivedC7 ⋊ C3×C7×C25
SylowC72 ⋊ C3×C25

Subgroups

subgroupG/H
center, hypercenterC7×C25ΓL(1,𝔽8)
commutator, hypocenterC7C3×C7×C25
FrattiniC5C5×C7×ΓL(1,𝔽8)
FittingC72×C25C3
socleC5×C72C3×C5
prime radicalC5×C7×ΓL(1,𝔽8)C5
cosocleC5×C7C3×C5×C7
LCSG ▷ C7
UCSC1 ◁ C7×C25

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 525

Conjugacy classes & character table

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

Relationships

maximal subgroup ofΓL(1,𝔽8)2×C25, C3×C7×ΓL(1,𝔽8)×C25, C7×C25×C22⋊ΓL(1,𝔽8)
maximal normal subgroup ofΓL(1,𝔽8)2×C25, C3×C7×ΓL(1,𝔽8)×C25
maximal quotient ofC7×C25×C7⋊C9, C3×C7×ΓL(1,𝔽8)×C25, C7×C25×C22⋊ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3675, 8);