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C25×ΓL(1,𝔽8)⋊D7 #20301226

order7350
isoclinismisoclinic class of ΓL(1,𝔽8)⋊D7
automorphism?
homologyAbelianization (H1) = C2×C3×C25 · Schur multiplier (H2) = C1

Aliases

direct productC25×ΓL(1,𝔽8)⋊D7
unambiguous semidirect productΓL(1,𝔽8)⋊(D7×C25)

Splittings

Fitting, HallC72×C25 ⋊ C2×C3
derived, SylowC72 ⋊ C2×C3×C25
HallC7×ΓL(1,𝔽8)×C25 ⋊ C2
HallC25×C7⋊D7 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ C2×C25
HallC7⋊D7 ⋊ C3×C25

Subgroups

subgroupG/H
center, hypercenterC25ΓL(1,𝔽8)⋊D7
commutator, hypocenterC72C2×C3×C25
FrattiniC5C5×ΓL(1,𝔽8)⋊D7
FittingC72×C25C2×C3
socleC5×C72C2×C3×C5
LCSG ▷ C72
UCSC1 ◁ C25

60 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

550 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×C25×AGL(1,𝔽7), (ΓL(1,𝔽8)⋊D7)⋊D25, D25×ΓL(1,𝔽8)⋊D7, …
maximal normal subgroup ofD7×C25×AGL(1,𝔽7), (ΓL(1,𝔽8)⋊D7)⋊D25, D25×ΓL(1,𝔽8)⋊D7, …
maximal quotient ofC25×ΓL(1,𝔽8)⋊Dic7, C2×C25×ΓL(1,𝔽8)⋊D7, C25×(C7⋊C9)⋊D7, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7350, 49);