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ΓL(1,𝔽8)⋊D13 #10587719

order546
isoclinismisoclinic class of ΓL(1,𝔽8)⋊D13
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊D13

Splittings

Fitting, socle, derived, HallC7×C13 ⋊ C2×C3
HallC13×ΓL(1,𝔽8) ⋊ C2
HallD91 ⋊ C3
HallΓL(1,𝔽8) ⋊ D13
SylowC13 ⋊ AGL(1,𝔽7)
SylowC7 ⋊ C3×D13

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C13C2×C3
LCSG ▷ C7×C13
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 91

Polycyclic presentation:

2,3,7,13|c^a=c^6,d^a=d^12,c^b=c^2

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD13×AGL(1,𝔽7), C2×ΓL(1,𝔽8)⋊D13, ΓL(1,𝔽8)⋊(D13⋊C3), …
maximal normal subgroup ofD13×AGL(1,𝔽7), C2×ΓL(1,𝔽8)⋊D13, ΓL(1,𝔽8)⋊(D13⋊C3), …
maximal quotient ofΓL(1,𝔽8)⋊Dic13, C2×ΓL(1,𝔽8)⋊D13, (C7⋊C9)⋊D13, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(546, 8);