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ΓL(1,𝔽8)×C3⋊#10596817 #49141480

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

Aliases

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

Splittings

Fitting, socle, derivedC3×C5×C7×C29 ⋊ C3×C4
HallC3×C5×ΓL(1,𝔽8)×C29 ⋊ C4
HallC3×ΓL(1,𝔽8)×C29 ⋊ Dic5
HallC5×C7×C29 ⋊ C3×Dic3
HallC3×C5×ΓL(1,𝔽8) ⋊ #1181
HallC7×C29 ⋊ C3×Dic15
HallC5×C29 ⋊ Dic3×ΓL(1,𝔽8)
HallC3×ΓL(1,𝔽8) ⋊ #10596817
HallC5×C7 ⋊ C3×C3⋊#1181
SylowC29 ⋊ ΓL(1,𝔽8)×Dic15
SylowC7 ⋊ C3×C3⋊#10596817
SylowC5 ⋊ ΓL(1,𝔽8)×C3⋊#1181

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C7×C29C3×C4
prime radicalC3×C5×ΓL(1,𝔽8)×D29C2
cosocleC3×C5×C7×D29C2×C3
LCSG ▷ C3×C5×C7×C29
UCSC1

112 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 39 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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36540, 322);