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C5×ΓL(1,𝔽8)⋊D37 #21441097

order7770
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×ΓL(1,𝔽8)⋊D37
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×D37)

Splittings

Fitting, socle, HallC5×C7×C37 ⋊ C2×C3
derived, HallC7×C37 ⋊ C2×C3×C5
HallC5×ΓL(1,𝔽8)×C37 ⋊ C2
HallC5×D259 ⋊ C3
HallΓL(1,𝔽8)×C37 ⋊ C2×C5
HallD259 ⋊ C3×C5
HallC5×C37 ⋊ AGL(1,𝔽7)
HallC5×ΓL(1,𝔽8) ⋊ D37
SylowC37 ⋊ C5×AGL(1,𝔽7)
HallC5×C7 ⋊ C3×D37
HallΓL(1,𝔽8) ⋊ C5×D37
SylowC7 ⋊ C3×C5×D37

Subgroups

subgroupG/H
center, hypercenterC5ΓL(1,𝔽8)⋊D37
commutator, hypocenterC7×C37C2×C3×C5
FrattiniC1G
Fitting, socleC5×C7×C37C2×C3
LCSG ▷ C7×C37
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

485 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×AGL(1,𝔽7)×D37, D5×ΓL(1,𝔽8)⋊D37, (C5⋊AGL(1,𝔽7))⋊D37, …
maximal normal subgroup ofC5×AGL(1,𝔽7)×D37, D5×ΓL(1,𝔽8)⋊D37, (C5⋊AGL(1,𝔽7))⋊D37, …
maximal quotient ofC5×ΓL(1,𝔽8)⋊Dic37, C2×C5×ΓL(1,𝔽8)⋊D37, C5×(C7⋊C9)⋊D37, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7770, 30);