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ΓL(1,𝔽8)×AGL(1,𝔽11) #14556350

order2310
isoclinismisoclinic class of ΓL(1,𝔽8)×AGL(1,𝔽11)
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×AGL(1,𝔽11)
unambiguous semidirect productD11⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product(C11⋊C5)⋊(C2×ΓL(1,𝔽8))
unambiguous semidirect product(ΓL(1,𝔽8)×D11)⋊C5
unambiguous semidirect product(C7×AGL(1,𝔽11))⋊C3

Splittings

Fitting, socle, derived, HallC7×C11 ⋊ C2×C3×C5
HallΓL(1,𝔽8)×C11⋊C5 ⋊ C2
HallC7×AGL(1,𝔽11) ⋊ C3
HallΓL(1,𝔽8)×D11 ⋊ C5
HallC7×C11⋊C5 ⋊ C2×C3
HallC11×ΓL(1,𝔽8) ⋊ C2×C5
HallC7×D11 ⋊ C3×C5
HallC11⋊C5 ⋊ C2×ΓL(1,𝔽8)
HallD11 ⋊ C5×ΓL(1,𝔽8)
SylowC11 ⋊ C2×C5×ΓL(1,𝔽8)
SylowC7 ⋊ C3×AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C11C2×C3×C5
LCSG ▷ C7×C11
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 77

A minimal faithful permutation representation — 2 generators on 18 points:

(2,3,4)(5,7,6)(9,13,14,12,17,10,11,18,15,16);(1,2,3,5,4,6,7)(8,9,12,17,11,14,18,13,16,15,10)

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×AGL(1,𝔽11), C2×ΓL(1,𝔽8)×AGL(1,𝔽11), C5×ΓL(1,𝔽8)×AGL(1,𝔽11), …
maximal normal subgroup ofAGL(1,𝔽7)×AGL(1,𝔽11), C2×ΓL(1,𝔽8)×AGL(1,𝔽11), C5×ΓL(1,𝔽8)×AGL(1,𝔽11), …
maximal quotient ofΓL(1,𝔽8)×Dic11⋊C5, C2×ΓL(1,𝔽8)×AGL(1,𝔽11), ΓL(1,𝔽8)×D11⋊C25, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2310, 34);