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ΓL(1,𝔽8)⋊D103 #16864504

order4326
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊D103

Splittings

Fitting, socle, derived, HallC7×C103 ⋊ C2×C3
HallΓL(1,𝔽8)×C103 ⋊ C2
HallD721 ⋊ C3
SylowC103 ⋊ AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ D103
SylowC7 ⋊ C3×D103

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C103C2×C3
LCSG ▷ C7×C103
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 110
transitive degreeμt(G) = 721

Polycyclic presentation:

2,3,7,103|c^a=c^6,d^a=d^102,c^b=c^2

Conjugacy classes & character table

262 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊(D103⋊C3), C3×ΓL(1,𝔽8)⋊D103, C3⋊(ΓL(1,𝔽8)⋊D103), …
maximal normal subgroup ofΓL(1,𝔽8)⋊(D103⋊C3), C3×ΓL(1,𝔽8)⋊D103, C5×ΓL(1,𝔽8)⋊D103, …
maximal quotient of(C7⋊C9)⋊D103, C3×ΓL(1,𝔽8)⋊D103, C3⋊(ΓL(1,𝔽8)⋊D103), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4326, 22);