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ΓL(1,𝔽8)⋊D79 #15693305

order3318
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊D79

Splittings

Fitting, socle, derived, HallC7×C79 ⋊ C2×C3
HallΓL(1,𝔽8)×C79 ⋊ C2
HallD553 ⋊ C3
SylowC79 ⋊ AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ D79
SylowC7 ⋊ C3×D79

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C79C2×C3
LCSG ▷ C7×C79
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 86
transitive degreeμt(G) = 553

Polycyclic presentation:

2,3,7,79|c^a=c^6,d^a=d^78,c^b=c^2

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×ΓL(1,𝔽8)⋊D79, C5⋊(ΓL(1,𝔽8)⋊D79), (C49⋊C3)⋊D79, …
maximal normal subgroup ofC5×ΓL(1,𝔽8)⋊D79, C7×ΓL(1,𝔽8)⋊D79, C11×ΓL(1,𝔽8)⋊D79
maximal quotient ofC5×ΓL(1,𝔽8)⋊D79, C5⋊(ΓL(1,𝔽8)⋊D79), (C49⋊C3)⋊D79, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3318, 22);