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C3×ΓL(1,𝔽8)⋊#10596817 #49141450

order36540
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)⋊#10596817
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×#10596817)
unambiguous semidirect product(C7⋊#10596817)⋊C32
unambiguous semidirect product(C3×C7⋊#10596817)⋊C3

Splittings

Fitting, socleC3×C5×C7×C29 ⋊ C3×C4
derived, HallC5×C7×C29 ⋊ C32×C4
HallC3×C5×ΓL(1,𝔽8)×C29 ⋊ C4
HallC7⋊#10596817 ⋊ C32
HallC3×ΓL(1,𝔽8)×C29 ⋊ Dic5
HallC3×C5×ΓL(1,𝔽8) ⋊ #1181
HallC7×C29 ⋊ C32×Dic5
HallC5×C29 ⋊ C3×ΓL(1,𝔽8)⋊C4
HallC3×ΓL(1,𝔽8) ⋊ #10596817
HallC5×C7 ⋊ C32×#1181
SylowC29 ⋊ C3×ΓL(1,𝔽8)⋊Dic5
SylowC7 ⋊ C32×#10596817
SylowC5 ⋊ C3×ΓL(1,𝔽8)⋊#1181

Subgroups

subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)⋊#10596817
commutator, hypocenterC5×C7×C29C32×C4
FrattiniC1G
Fitting, socleC3×C5×C7×C29C3×C4
prime radicalC3×C5×ΓL(1,𝔽8)×D29C2
cosocleC5×C7×D29C2×C32
LCSG ▷ C5×C7×C29
UCSC1 ◁ C3

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3045

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 292);