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Cayley diagram

ΓL(1,𝔽8)×D11 #90755

order462
isoclinismisoclinic class of ΓL(1,𝔽8)×D11
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D11
unambiguous semidirect productC11⋊(C2×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D11)⋊C3

Splittings

Fitting, socle, derived, HallC7×C11 ⋊ C2×C3
HallC11×ΓL(1,𝔽8) ⋊ C2
HallC7×D11 ⋊ C3
SylowC11 ⋊ C2×ΓL(1,𝔽8)
SylowC7 ⋊ C3×D11

Subgroups

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

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD11×AGL(1,𝔽7), C2×ΓL(1,𝔽8)×D11, C3×ΓL(1,𝔽8)×D11, …
maximal normal subgroup ofD11×AGL(1,𝔽7), C2×ΓL(1,𝔽8)×D11, C3×ΓL(1,𝔽8)×D11, …
maximal quotient ofΓL(1,𝔽8)×Dic11, C2×ΓL(1,𝔽8)×D11, D11×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(462, 1);