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C2×AGL(1,𝔽11)×#3861 #44568574

order32340
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×AGL(1,𝔽11)×#3861
unambiguous semidirect productD11⋊(C2×C5×#3861)
unambiguous semidirect product(C2×D11)⋊(C5×#3861)
unambiguous semidirect product(C11⋊C5)⋊(C22×#3861)
unambiguous semidirect product(C2×C11⋊C5)⋊(C2×#3861)
unambiguous semidirect product(C2×D11×#3861)⋊C5

Splittings

Fitting, socleC2×C72×C11 ⋊ C2×C3×C5
derived, HallC72×C11 ⋊ C22×C3×C5
HallC2×C72×AGL(1,𝔽11) ⋊ C3
HallC11⋊C5×#3861 ⋊ C22
HallC2×D11×#3861 ⋊ C5
HallC72×C11⋊C5 ⋊ C22×C3
HallC2×C72×D11 ⋊ C3×C5
HallC11×#3861 ⋊ C22×C5
HallC11⋊C5 ⋊ C22×#3861
SylowC72 ⋊ C2×C3×AGL(1,𝔽11)
HallC2×D11 ⋊ C5×#3861
SylowC11 ⋊ C22×C5×#3861

Subgroups

subgroupG/H
center, hypercenterC2AGL(1,𝔽11)×#3861
commutator, cosocle, hypocenterC72×C11C22×C3×C5
FrattiniC1G
Fitting, socleC2×C72×C11C2×C3×C5
LCSG ▷ C72×C11
UCSC1 ◁ C2

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 1078

A minimal faithful permutation representation — 3 generators on 27 points:

(2,6,7,5,10,3,4,11,8,9)(13,14,15)(16,18,17)(20,21,22)(23,25,24);(1,2,5,10,4,7,11,6,9,8,3)(19,20,21,23,22,24,25)(26,27);(12,13,14,16,15,17,18)

Conjugacy classes & character table

418 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 276);