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C5×C5⋊(ΓL(1,𝔽8)⋊D7) #20301267

order7350
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C5⋊(ΓL(1,𝔽8)⋊D7)
unambiguous semidirect productC5⋊(C5×ΓL(1,𝔽8)⋊D7)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×D35)

Splittings

Fitting, socle, HallC52×C72 ⋊ C2×C3
derivedC5×C72 ⋊ C2×C3×C5
HallC52×C7×ΓL(1,𝔽8) ⋊ C2
HallC5×C5⋊(C7⋊D7) ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ C5×D5
SylowC72 ⋊ C3×C5×D5
SylowC52 ⋊ ΓL(1,𝔽8)⋊D7

Subgroups

subgroupG/H
center, hypercenterC5C5⋊(ΓL(1,𝔽8)⋊D7)
commutator, hypocenterC5×C72C2×C3×C5
FrattiniC1G
Fitting, socleC52×C72C2×C3
LCSG ▷ C5×C72
UCSC1 ◁ C5

100 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

460 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×AGL(1,𝔽7)×D35, C5×D7×C5⋊AGL(1,𝔽7), C5×D5×ΓL(1,𝔽8)⋊D7, …
maximal normal subgroup ofC5×AGL(1,𝔽7)×D35, C5×D7×C5⋊AGL(1,𝔽7), C5×D5×ΓL(1,𝔽8)⋊D7, …
maximal quotient ofC5×ΓL(1,𝔽8)⋊Dic35, C2×C5×C5⋊(ΓL(1,𝔽8)⋊D7), C5×C5⋊((C7⋊C9)⋊D7), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7350, 90);