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ΓL(1,𝔽8)⋊D83 #16812552

order3486
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊D83
unambiguous semidirect productC83⋊AGL(1,𝔽7)
unambiguous semidirect productD581⋊C3

Splittings

Fitting, socle, derived, HallC7×C83 ⋊ C2×C3
HallΓL(1,𝔽8)×C83 ⋊ C2
HallD581 ⋊ C3
SylowC83 ⋊ AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ D83
SylowC7 ⋊ C3×D83

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C83C2×C3
LCSG ▷ C7×C83
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 90
transitive degreeμt(G) = 581

Polycyclic presentation:

2,3,7,83|c^a=c^6,d^a=d^82,c^b=c^2

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×ΓL(1,𝔽8)⋊D83, C5⋊(ΓL(1,𝔽8)⋊D83), C83⋊(D49⋊C3), …
maximal normal subgroup ofC5×ΓL(1,𝔽8)⋊D83, C7×ΓL(1,𝔽8)⋊D83, C11×ΓL(1,𝔽8)⋊D83
maximal quotient ofC5×ΓL(1,𝔽8)⋊D83, C5⋊(ΓL(1,𝔽8)⋊D83), C83⋊(D49⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3486, 12);