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ΓL(1,𝔽8)×C22⋊D11 #14270441

order1848
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productΓL(1,𝔽8)×C22⋊D11
unambiguous semidirect product(C7×C22⋊D11)⋊C3

Splittings

FittingC22×C7×C11 ⋊ C2×C3
HallC7×C22⋊D11 ⋊ C3
HallC11×ΓL(1,𝔽8) ⋊ D4
HallC7×C11 ⋊ C3×D4
SylowC11 ⋊ D4×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C22⋊D11

Subgroups

subgroupG/H
center, FrattiniC2C2×ΓL(1,𝔽8)×D11
commutator, socle, cosocleC2×C7×C11C22×C3
FittingC22×C7×C11C2×C3
hypercenterC22ΓL(1,𝔽8)×D11
hypocenterC7×C11C3×D4
LCSG ▷ C2×C7×C11 ▷ C7×C11
UCSC1 ◁ C2 ◁ C22

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 308

A minimal faithful permutation representation — 2 generators on 22 points:

(2,3,4)(5,7,6)(8,9,11,13,15,17,18,16,14,12,10)(20,22);(1,2,3,5,4,6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1848, 24);