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ΓL(1,𝔽8)⋊(C71⋊AGL(1,𝔽5)) #44447203

order29820
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊(C71⋊AGL(1,𝔽5))
unambiguous semidirect productC71⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽5))
unambiguous semidirect product(C7⋊(C71⋊AGL(1,𝔽5)))⋊C3

Splittings

Fitting, socle, derived, HallC5×C7×C71 ⋊ C3×C4
HallC7⋊(C71⋊AGL(1,𝔽5)) ⋊ C3
HallC5×ΓL(1,𝔽8)×C71 ⋊ C4
HallΓL(1,𝔽8)×C71 ⋊ AGL(1,𝔽5)
HallC7×C71 ⋊ C3×AGL(1,𝔽5)
HallC5×C71 ⋊ ΓL(1,𝔽8)⋊C4
HallC5×ΓL(1,𝔽8) ⋊ Dic71
SylowC71 ⋊ ΓL(1,𝔽8)⋊AGL(1,𝔽5)
HallC5×C7 ⋊ C3×Dic71
HallΓL(1,𝔽8) ⋊ C71⋊AGL(1,𝔽5)
SylowC7 ⋊ C3×C71⋊AGL(1,𝔽5)
SylowC5 ⋊ ΓL(1,𝔽8)⋊Dic71

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C7×C71C3×C4
prime radicalD5×ΓL(1,𝔽8)×C71C2
cosocleC7×D5×C71C2×C3
LCSG ▷ C5×C7×C71
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2485

Polycyclic presentation:

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

Conjugacy classes & character table

719 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29820, 91);