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ΓL(1,𝔽8)×C29×ΓL(1,𝔽16) #49141470

order36540
automorphism?
homologyAbelianization (H1) = C3×C4×C29 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×C29×ΓL(1,𝔽16)
unambiguous semidirect productC3⋊(AGL(1,𝔽5)×ΓL(1,𝔽8)×C29)
unambiguous semidirect product(C7×ΓL(1,𝔽16))⋊(C3×C29)
unambiguous semidirect product(C7×C29×ΓL(1,𝔽16))⋊C3

Splittings

Fitting, socleC3×C5×C7×C29 ⋊ C3×C4
derivedC3×C5×C7 ⋊ C3×C4×C29
HallC3×C5×ΓL(1,𝔽8)×C29 ⋊ C4
HallC3×ΓL(1,𝔽8)×C29 ⋊ AGL(1,𝔽5)
HallC5×C7×C29 ⋊ C3×Dic3
HallC3×C5×ΓL(1,𝔽8) ⋊ C4×C29
HallC7×C29 ⋊ C3×ΓL(1,𝔽16)
HallC5×C29 ⋊ Dic3×ΓL(1,𝔽8)
HallC3×ΓL(1,𝔽8) ⋊ AGL(1,𝔽5)×C29
HallC5×C7 ⋊ C3×Dic3×C29
SylowC7 ⋊ C3×C29×ΓL(1,𝔽16)
SylowC5 ⋊ Dic3×ΓL(1,𝔽8)×C29

Subgroups

subgroupG/H
center, hypercenterC29ΓL(1,𝔽8)×ΓL(1,𝔽16)
commutator, hypocenterC3×C5×C7C3×C4×C29
FrattiniC1G
Fitting, socleC3×C5×C7×C29C3×C4
prime radicalC3×D5×ΓL(1,𝔽8)×C29C2
cosocleC3×C7×D5C2×C3×C29
LCSG ▷ C3×C5×C7
UCSC1 ◁ C29

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

normal subgroups only

This group has 112 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) = 3045

Conjugacy classes & character table

1305 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 312);