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C3⋊((C43⋊C7)⋊AGL(1,𝔽5)) #31825396

order18060
automorphism?
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊((C43⋊C7)⋊AGL(1,𝔽5))
unambiguous semidirect product(C43⋊C7)⋊ΓL(1,𝔽16)
unambiguous semidirect product(C3⋊(C43⋊AGL(1,𝔽5)))⋊C7

Splittings

Fitting, socle, derived, HallC3×C5×C43 ⋊ C4×C7
HallC3×C5×C43⋊C7 ⋊ C4
HallC3⋊(C43⋊AGL(1,𝔽5)) ⋊ C7
HallC5×C43⋊C7 ⋊ Dic3
HallC3×C43⋊C7 ⋊ AGL(1,𝔽5)
HallC43⋊C7 ⋊ ΓL(1,𝔽16)
HallC5×C43 ⋊ C7×Dic3
HallC3×C43 ⋊ C7×AGL(1,𝔽5)
SylowC43 ⋊ C7×ΓL(1,𝔽16)
HallC3×C5 ⋊ Dic43⋊C7
SylowC5 ⋊ C3⋊(Dic43⋊C7)
SylowC3 ⋊ (C43⋊C7)⋊AGL(1,𝔽5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C43C4×C7
prime radicalC3×D5×C43⋊C7C2
cosocleC3×D5×C43C2×C7
LCSG ▷ C3×C5×C43
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 645

Polycyclic presentation:

2,7,2,3,5,43|a^2=c,d^a=d^2,e^a=e^2,f^a=f^42,f^b=f^41,e^c=e^4

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18060, 72);