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C3⋊(C7⋊(C71⋊AGL(1,𝔽5))) #44447248

order29820
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C7⋊(C71⋊AGL(1,𝔽5)))
unambiguous semidirect productC7⋊(C3⋊(C71⋊AGL(1,𝔽5)))
unambiguous semidirect productC71⋊(C3⋊(C7⋊AGL(1,𝔽5)))

Splittings

Fitting, socle, derived, HallC3×C5×C7×C71 ⋊ C4
HallC5×C7×C71 ⋊ Dic3
HallC3×C7×C71 ⋊ AGL(1,𝔽5)
HallC3×C5×C71 ⋊ Dic7
HallC7×C71 ⋊ ΓL(1,𝔽16)
HallC5×C71 ⋊ Dic21
HallC3×C71 ⋊ C7⋊AGL(1,𝔽5)
HallC3×C5×C7 ⋊ Dic71
SylowC71 ⋊ C3⋊(C7⋊AGL(1,𝔽5))
HallC5×C7 ⋊ Dic213
HallC3×C7 ⋊ C71⋊AGL(1,𝔽5)
HallC3×C5 ⋊ Dic497
SylowC7 ⋊ C3⋊(C71⋊AGL(1,𝔽5))
SylowC5 ⋊ Dic1491
SylowC3 ⋊ C7⋊(C71⋊AGL(1,𝔽5))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C7×C71C4
prime radical, cosocleC3×C7×D5×C71C2
LCSG ▷ C3×C5×C7×C71
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 7455

Polycyclic presentation:

2^2,3,5,7,71|a^2=b,c^a=c^2,d^a=d^2,e^a=e^6,f^a=f^70,d^b=d^4

Conjugacy classes & character table

2985 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29820, 136);