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C3⋊(C7⋊(C7⋊(C11⋊AGL(1,𝔽5)))) #44568864

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

Aliases

unambiguous semidirect productC3⋊(C7⋊(C7⋊(C11⋊AGL(1,𝔽5))))
unambiguous semidirect productC7⋊(C3⋊(C7⋊(C11⋊AGL(1,𝔽5))))
unambiguous semidirect productC11⋊(C3⋊(C7⋊(C7⋊AGL(1,𝔽5))))

Splittings

Fitting, socle, derived, HallC3×C5×C72×C11 ⋊ C4
HallC5×C72×C11 ⋊ Dic3
HallC3×C72×C11 ⋊ AGL(1,𝔽5)
HallC3×C5×C72 ⋊ Dic11
HallC72×C11 ⋊ ΓL(1,𝔽16)
HallC5×C72 ⋊ Dic33
HallC3×C5×C11 ⋊ C7⋊Dic7
HallC3×C72 ⋊ C11⋊AGL(1,𝔽5)
HallC5×C11 ⋊ C3⋊(C7⋊Dic7)
SylowC72 ⋊ C3⋊(C11⋊AGL(1,𝔽5))
HallC3×C11 ⋊ C7⋊(C7⋊AGL(1,𝔽5))
HallC3×C5 ⋊ C7⋊Dic77
SylowC11 ⋊ C3⋊(C7⋊(C7⋊AGL(1,𝔽5)))
SylowC5 ⋊ C3⋊(C7⋊Dic77)
SylowC3 ⋊ C7⋊(C7⋊(C11⋊AGL(1,𝔽5)))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C72×C11C4
prime radical, cosocleC3×C72×D5×C11C2
LCSG ▷ C3×C5×C72×C11
UCSC1

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 8085

Polycyclic presentation:

2^2,3,5,7^2,11|a^2=b,c^a=c^2,d^a=d^2,e^a=e^6,f^a=f^6,g^a=g^10,d^b=d^4

Conjugacy classes & character table

3237 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 566);