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ΓL(1,𝔽8)⋊(Dic71⋊C5) #44447160

order29820
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊(Dic71⋊C5)
unambiguous semidirect product(C71⋊C5)⋊(ΓL(1,𝔽8)⋊C4)
unambiguous semidirect product(ΓL(1,𝔽8)⋊Dic71)⋊C5
unambiguous semidirect product(C7⋊(Dic71⋊C5))⋊C3

Splittings

derived, HallC7×C71 ⋊ C3×C4×C5
HallC7⋊(Dic71⋊C5) ⋊ C3
HallΓL(1,𝔽8)×C71⋊C5 ⋊ C4
HallΓL(1,𝔽8)⋊Dic71 ⋊ C5
HallC7×C71⋊C5 ⋊ C3×C4
HallDic497 ⋊ C3×C5
HallΓL(1,𝔽8)×C71 ⋊ C4×C5
HallC71⋊C5 ⋊ ΓL(1,𝔽8)⋊C4
SylowC71 ⋊ C5×ΓL(1,𝔽8)⋊C4
HallΓL(1,𝔽8) ⋊ Dic71⋊C5
SylowC7 ⋊ C3×Dic71⋊C5

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)⋊(D71⋊C5)
commutator, hypocenterC7×C71C3×C4×C5
Fitting, socle, cosocleC2×C7×C71C2×C3×C5
prime radicalC2×ΓL(1,𝔽8)×C71⋊C5C2
LCSG ▷ C7×C71
UCSC1 ◁ C2

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1988

Polycyclic presentation:

2,3,5,7,71,2|a^2=f,d^a=d^6,e^a=e^70,d^b=d^2,e^c=e^54

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29820, 48);