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C7×ΓL(1,𝔽8)×#63120 #50303847

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

Aliases

direct productC7×ΓL(1,𝔽8)×#63120

Splittings

Fitting, socle, HallC5×C72×C13 ⋊ C3×C4
derivedC5×C7×C13 ⋊ C3×C4×C7
HallC72×#63120 ⋊ C3
HallC5×C7×C13×ΓL(1,𝔽8) ⋊ C4
HallC7×C13×ΓL(1,𝔽8) ⋊ AGL(1,𝔽5)
HallC5×C7×ΓL(1,𝔽8) ⋊ Dic13
HallC72×C13 ⋊ C3×AGL(1,𝔽5)
HallC5×C72 ⋊ C3×Dic13
HallC5×C13 ⋊ C4×C7×ΓL(1,𝔽8)
SylowC72 ⋊ C3×#63120
SylowC13 ⋊ C7×AGL(1,𝔽5)×ΓL(1,𝔽8)
SylowC5 ⋊ C7×ΓL(1,𝔽8)×Dic13

Subgroups

subgroupG/H
center, hypercenterC7ΓL(1,𝔽8)×#63120
commutator, hypocenterC5×C7×C13C3×C4×C7
FrattiniC1G
Fitting, socleC5×C72×C13C3×C4
prime radicalC7×D5×C13×ΓL(1,𝔽8)C2
cosocleC7×D5×C13C2×C3×C7
LCSG ▷ C5×C7×C13
UCSC1 ◁ C7

120 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 1365

Conjugacy classes & character table

1015 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38220, 533);