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C5×ΓL(1,𝔽8)×C61 #20264288

order6405
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C5×C61 · Schur multiplier (H2) = C1

Aliases

direct productC5×ΓL(1,𝔽8)×C61
unambiguous semidirect productC7⋊(C3×C5×C61)
unambiguous semidirect product(C5×C7)⋊(C3×C61)

Splittings

Fitting, socle, HallC5×C7×C61 ⋊ C3
derived, SylowC7 ⋊ C3×C5×C61
HallC7×C61 ⋊ C3×C5
HallC5×C7 ⋊ C3×C61

Subgroups

subgroupG/H
center, hypercenterC5×C61ΓL(1,𝔽8)
commutator, hypocenterC7C3×C5×C61
FrattiniC1G
Fitting, socleC5×C7×C61C3
LCSG ▷ C7
UCSC1 ◁ C5×C61

16 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1525 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×C5×ΓL(1,𝔽8)×C61, C5×ΓL(1,𝔽8)×D61, C5×AGL(1,𝔽7)×C61, …
maximal normal subgroup ofC2×C5×ΓL(1,𝔽8)×C61, C5×ΓL(1,𝔽8)×D61, C5×AGL(1,𝔽7)×C61, …
maximal quotient ofC2×C5×ΓL(1,𝔽8)×C61, C5×C61×C7⋊C9, C3×C5×ΓL(1,𝔽8)×C61, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6405, 4);