Finite Group Databasebeta

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C5×C3⋊(ΓL(1,𝔽8)⋊#1181) #49141458

order36540
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C3⋊(ΓL(1,𝔽8)⋊#1181)
unambiguous semidirect productC3⋊(C5×ΓL(1,𝔽8)⋊#1181)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×C3⋊#1181)
unambiguous semidirect product(C3⋊(C7⋊#1181))⋊(C3×C5)
unambiguous semidirect product(C5×C3⋊(C7⋊#1181))⋊C3

Splittings

Fitting, socleC3×C5×C7×C29 ⋊ C3×C4
derivedC3×C7×C29 ⋊ C3×C4×C5
HallC3×C5×ΓL(1,𝔽8)×C29 ⋊ C4
HallC3×ΓL(1,𝔽8)×C29 ⋊ C4×C5
HallC5×C7×C29 ⋊ C3×Dic3
HallC3×C5×ΓL(1,𝔽8) ⋊ #1181
HallC7×C29 ⋊ C3×C5×Dic3
HallC5×C29 ⋊ C3⋊(ΓL(1,𝔽8)⋊C4)
HallC3×ΓL(1,𝔽8) ⋊ C5×#1181
HallC5×C7 ⋊ C3×C3⋊#1181
SylowC29 ⋊ C5×C3⋊(ΓL(1,𝔽8)⋊C4)
SylowC7 ⋊ C3×C5×C3⋊#1181

Subgroups

subgroupG/H
center, hypercenterC5C3⋊(ΓL(1,𝔽8)⋊#1181)
commutator, hypocenterC3×C7×C29C3×C4×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C29C3×C4
prime radicalC3×C5×ΓL(1,𝔽8)×D29C2
cosocleC3×C7×D29C2×C3×C5
LCSG ▷ C3×C7×C29
UCSC1 ◁ C5

112 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 3045

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 300);