Finite Group Databasebeta

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C72×C11×ΓL(1,𝔽16) #44568859

order32340
automorphism?
homologyAbelianization (H1) = C4×C72×C11 · Schur multiplier (H2) = C7

Aliases

direct productC72×C11×ΓL(1,𝔽16)
unambiguous semidirect productC3⋊(C72×C11×AGL(1,𝔽5))

Splittings

Fitting, socle, HallC3×C5×C72×C11 ⋊ C4
derived, HallC3×C5 ⋊ C4×C72×C11
HallC5×C72×C11 ⋊ Dic3
HallC3×C72×C11 ⋊ AGL(1,𝔽5)
HallC3×C5×C72 ⋊ C4×C11
HallC5×C72 ⋊ C11×Dic3
HallC3×C5×C11 ⋊ C4×C72
HallC3×C72 ⋊ C11×AGL(1,𝔽5)
HallC5×C11 ⋊ C72×Dic3
HallC3×C11 ⋊ C72×AGL(1,𝔽5)
SylowC5 ⋊ C72×C11×Dic3
SylowC3 ⋊ C72×C11×AGL(1,𝔽5)

Subgroups

subgroupG/H
center, hypercenterC72×C11ΓL(1,𝔽16)
commutator, hypocenterC3×C5C4×C72×C11
FrattiniC1G
Fitting, socleC3×C5×C72×C11C4
prime radicalC3×C72×D5×C11C2
cosocleC3×D5C2×C72×C11
LCSG ▷ C3×C5
UCSC1 ◁ C72×C11

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 8085

Conjugacy classes & character table

4851 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 561);