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C2×C3⋊(C7⋊(C7⋊AGL(1,𝔽11))) #44568758

order32340
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2×C7

Aliases

direct productC2×C3⋊(C7⋊(C7⋊AGL(1,𝔽11)))
unambiguous semidirect product(C2×C3)⋊(C7⋊(C7⋊AGL(1,𝔽11)))
unambiguous semidirect product(C2×C7)⋊(C3⋊(C7⋊AGL(1,𝔽11)))
unambiguous semidirect product(C2×C11⋊C5)⋊(C3⋊(C7⋊D7))
unambiguous semidirect product(C2×C3⋊(C7⋊D77))⋊C5

Splittings

Fitting, socleC2×C3×C72×C11 ⋊ C2×C5
derived, HallC3×C72×C11 ⋊ C22×C5
HallC3×C72×C11⋊C5 ⋊ C22
HallC2×C3⋊(C7⋊D77) ⋊ C5
HallC72×C11⋊C5 ⋊ C2×D3
HallC72×C11 ⋊ C2×C5×D3
HallC3×C11⋊C5 ⋊ C2×C7⋊D7
HallC3×C72 ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ C2×C3⋊(C7⋊D7)
SylowC72 ⋊ C2×C3⋊AGL(1,𝔽11)
HallC3×C11 ⋊ C2×C5×C7⋊D7
SylowC11 ⋊ C2×C5×C3⋊(C7⋊D7)
SylowC3 ⋊ C2×C7⋊(C7⋊AGL(1,𝔽11))

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(C7⋊(C7⋊AGL(1,𝔽11)))
commutator, cosocle, hypocenterC3×C72×C11C22×C5
FrattiniC1G
Fitting, socleC2×C3×C72×C11C2×C5
LCSG ▷ C3×C72×C11
UCSC1 ◁ C2

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 3234

A minimal faithful permutation representation — 3 generators on 30 points:

(2,6,7,5,10,3,4,11,8,9)(13,14)(16,17)(18,19)(20,21)(23,24)(25,26)(27,28);(1,2,5,10,4,7,11,6,9,8,3)(12,13,14)(22,23,25,27,28,26,24)(29,30);(15,16,18,20,21,19,17)

Conjugacy classes & character table

1044 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 460);