Finite Group Databasebeta

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C4×C5×C7×C13×ΓL(1,𝔽8) #50303675

order38220
automorphism?
homologyAbelianization (H1) = C3×C4×C5×C7×C13 · Schur multiplier (H2) = C1

Aliases

direct productC4×C5×C7×C13×ΓL(1,𝔽8)
unambiguous semidirect product(C4×C7)⋊(C3×C5×C7×C13)
unambiguous semidirect product(C4×C5×C7)⋊(C3×C7×C13)

Splittings

Fitting, HallC4×C5×C72×C13 ⋊ C3
derivedC7 ⋊ C3×C4×C5×C7×C13
HallC5×C72×C13 ⋊ C3×C4
HallC4×C72×C13 ⋊ C3×C5
HallC4×C5×C72 ⋊ C3×C13
HallC72×C13 ⋊ C3×C4×C5
HallC5×C72 ⋊ C3×C4×C13
HallC4×C72 ⋊ C3×C5×C13
SylowC72 ⋊ C3×C4×C5×C13

Subgroups

subgroupG/H
center, hypercenterC4×C5×C7×C13ΓL(1,𝔽8)
commutator, hypocenterC7C3×C4×C5×C7×C13
FrattiniC2C2×C5×C7×C13×ΓL(1,𝔽8)
FittingC4×C5×C72×C13C3
socleC2×C5×C72×C13C2×C3
prime radicalC2×C5×C7×C13×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C3×C5×C7×C13
LCSG ▷ C7
UCSC1 ◁ C4×C5×C7×C13

120 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 5460

Conjugacy classes & character table

9100 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38220, 361);