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Cayley diagram

C2×C11×ΓL(1,𝔽8) #90758

order462
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) = C2×C3×C11 · Schur multiplier (H2) = C1

Aliases

direct productC2×C11×ΓL(1,𝔽8)
unambiguous semidirect product(C2×C7)⋊(C3×C11)
unambiguous semidirect product(C2×C7×C11)⋊C3

Splittings

Fitting, socle, HallC2×C7×C11 ⋊ C3
derived, SylowC7 ⋊ C2×C3×C11
HallC7×C11 ⋊ C2×C3
HallC2×C7 ⋊ C3×C11

Subgroups

subgroupG/H
center, hypercenterC2×C11ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C3×C11
FrattiniC1G
Fitting, socleC2×C7×C11C3
LCSG ▷ C7
UCSC1 ◁ C2×C11

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 154

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2)(4,5,6)(7,9,8)(10,11,12,13,14,15,16,17,18,19,20);(3,4,5,7,6,8,9)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×Dic11, C11×ΓL(1,𝔽8)⋊C4, C11⋊(ΓL(1,𝔽8)⋊C4), …
maximal normal subgroup ofΓL(1,𝔽8)×Dic11, C11×ΓL(1,𝔽8)⋊C4, C11⋊(ΓL(1,𝔽8)⋊C4), …
maximal quotient ofC4×C11×ΓL(1,𝔽8), C22×C11×ΓL(1,𝔽8), C2×C11×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(462, 4);