Finite Group Databasebeta

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C4×C5×C7×C11×ΓL(1,𝔽8) #44568514

order32340
automorphism?
homologyAbelianization (H1) = C3×C4×C5×C7×C11 · Schur multiplier (H2) = C1

Aliases

direct productC4×C5×C7×C11×ΓL(1,𝔽8)
unambiguous semidirect product(C4×C7)⋊(C3×C5×C7×C11)
unambiguous semidirect product(C4×C5×C7)⋊(C3×C7×C11)
unambiguous semidirect product(C4×C5×C7×C11)⋊(C3×C7)

Splittings

Fitting, HallC4×C5×C72×C11 ⋊ C3
derivedC7 ⋊ C3×C4×C5×C7×C11
HallC5×C72×C11 ⋊ C3×C4
HallC4×C72×C11 ⋊ C3×C5
HallC4×C5×C72 ⋊ C3×C11
HallC72×C11 ⋊ C3×C4×C5
HallC5×C72 ⋊ C3×C4×C11
HallC4×C72 ⋊ C3×C5×C11
SylowC72 ⋊ C3×C4×C5×C11

Subgroups

subgroupG/H
center, hypercenterC4×C5×C7×C11ΓL(1,𝔽8)
commutator, hypocenterC7C3×C4×C5×C7×C11
FrattiniC2C2×C5×C7×C11×ΓL(1,𝔽8)
FittingC4×C5×C72×C11C3
socleC2×C5×C72×C11C2×C3
prime radicalC2×C5×C7×C11×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C3×C5×C7×C11
LCSG ▷ C7
UCSC1 ◁ C4×C5×C7×C11

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) = 4620

Conjugacy classes & character table

7700 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 216);