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C52×ΓL(1,𝔽8)⋊D7 #20301258

order7350
isoclinismisoclinic class of ΓL(1,𝔽8)⋊D7
automorphism?
homologyAbelianization (H1) = C2×C3×C52 · Schur multiplier (H2) = C5

Aliases

direct productC52×ΓL(1,𝔽8)⋊D7
unambiguous semidirect productΓL(1,𝔽8)⋊(C52×D7)

Splittings

Fitting, socle, HallC52×C72 ⋊ C2×C3
derived, SylowC72 ⋊ C2×C3×C52
HallC52×C7×ΓL(1,𝔽8) ⋊ C2
HallC52×C7⋊D7 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ C2×C52
HallC7⋊D7 ⋊ C3×C52

Subgroups

subgroupG/H
center, hypercenterC52ΓL(1,𝔽8)⋊D7
commutator, hypocenterC72C2×C3×C52
FrattiniC1G
Fitting, socleC52×C72C2×C3
LCSG ▷ C72
UCSC1 ◁ C52

160 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

550 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC52×D7×AGL(1,𝔽7), C5×(C5⋊AGL(1,𝔽7))⋊D7, C5×D5×ΓL(1,𝔽8)⋊D7, …
maximal normal subgroup ofC52×D7×AGL(1,𝔽7), C5×(C5⋊AGL(1,𝔽7))⋊D7, C5×D5×ΓL(1,𝔽8)⋊D7, …
maximal quotient ofC52×ΓL(1,𝔽8)⋊Dic7, C2×C52×ΓL(1,𝔽8)⋊D7, C52×(C7⋊C9)⋊D7, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7350, 81);