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Cayley diagram

C11×ΓL(1,𝔽8) #6595

order231
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphismAut(G) = C2×C5×AGL(1,𝔽7) (order 420) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C22×C5
homologyAbelianization (H1) = C3×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×ΓL(1,𝔽8)
unambiguous semidirect productC7⋊(C3×C11)
unambiguous semidirect product(C7×C11)⋊C3

Splittings

Fitting, socle, HallC7×C11 ⋊ C3
derived, SylowC7 ⋊ C3×C11

Subgroups

subgroupG/H
center, hypercenterC11ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C3×C11
FrattiniC1G
Fitting, socleC7×C11C3
LCSG ▷ C7
UCSC1 ◁ C11

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×D11, C11×AGL(1,𝔽7), C11⋊AGL(1,𝔽7), …
maximal normal subgroup ofΓL(1,𝔽8)×D11, C11×AGL(1,𝔽7), C11⋊AGL(1,𝔽7), …
maximal quotient ofC2×C11×ΓL(1,𝔽8), C11×C7⋊C9, C3×C11×ΓL(1,𝔽8), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(231, 1);