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C7×ΓL(1,𝔽8)⋊#63122 #50303855

order38220
automorphism?
homologyAbelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×ΓL(1,𝔽8)⋊#63122
unambiguous semidirect productΓL(1,𝔽8)⋊(C7×#63122)

Splittings

Fitting, socle, HallC5×C72×C13 ⋊ C3×C4
derivedC5×C7×C13 ⋊ C3×C4×C7
HallC7×C7⋊#63122 ⋊ C3
HallC5×C7×C13×ΓL(1,𝔽8) ⋊ C4
HallC7×C13×ΓL(1,𝔽8) ⋊ AGL(1,𝔽5)
HallC5×C7×ΓL(1,𝔽8) ⋊ #261
HallC72×C13 ⋊ C3×AGL(1,𝔽5)
HallC5×C72 ⋊ C3×#261
HallC7×ΓL(1,𝔽8) ⋊ #63122
HallC5×C13 ⋊ C7×ΓL(1,𝔽8)⋊C4
SylowC72 ⋊ C3×#63122
SylowC13 ⋊ C7×ΓL(1,𝔽8)⋊AGL(1,𝔽5)
SylowC5 ⋊ C7×ΓL(1,𝔽8)⋊#261

Subgroups

subgroupG/H
center, hypercenterC7ΓL(1,𝔽8)⋊#63122
commutator, hypocenterC5×C7×C13C3×C4×C7
FrattiniC1G
Fitting, socleC5×C72×C13C3×C4
prime radicalC7×ΓL(1,𝔽8)×D65C2
cosocleC7×D65C2×C3×C7
LCSG ▷ C5×C7×C13
UCSC1 ◁ C7

104 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2730

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38220, 541);