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C2×(C11⋊C5)⋊(D49⋊C3) #44568337

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

Aliases

direct productC2×(C11⋊C5)⋊(D49⋊C3)
unambiguous semidirect product(C2×C11⋊C5)⋊(D49⋊C3)
unambiguous semidirect product(C2×C49⋊C3)⋊AGL(1,𝔽11)
unambiguous semidirect product(C2×C11⋊(D49⋊C3))⋊C5
unambiguous semidirect product(C2×C49⋊AGL(1,𝔽11))⋊C3

Splittings

FittingC2×C11×C49 ⋊ C2×C3×C5
derived, HallC11×C49 ⋊ C22×C3×C5
HallC2×C49⋊AGL(1,𝔽11) ⋊ C3
HallC11⋊C5×C49⋊C3 ⋊ C22
HallC2×C11⋊(D49⋊C3) ⋊ C5
HallC49×C11⋊C5 ⋊ C22×C3
HallC2×D539 ⋊ C3×C5
HallC11×C49⋊C3 ⋊ C22×C5
HallC49⋊C3 ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ C2×D49⋊C3
SylowC49 ⋊ C2×C3×AGL(1,𝔽11)
SylowC11 ⋊ C2×C5×D49⋊C3

Subgroups

subgroupG/H
center, hypercenterC2(C11⋊C5)⋊(D49⋊C3)
commutator, cosocle, hypocenterC11×C49C22×C3×C5
FrattiniC7C2×ΓL(1,𝔽8)⋊AGL(1,𝔽11)
FittingC2×C11×C49C2×C3×C5
socleC2×C7×C11C5×AGL(1,𝔽7)
LCSG ▷ C11×C49
UCSC1 ◁ C2

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 62
transitive degreeμt(G) = 1078

A minimal faithful permutation representation — 2 generators on 62 points:

(2,6,7,5,10,3,4,11,8,9)(13,21,23,15,16,24)(14,33,35,19,20,36)(17,45,32,26,27,54)(18,50,51,30,31,60)(22,29,28,34,39,47)(25,48,57,43,44,58)(37,49,52,55,38,41)(40,53,42,56,46,59);(1,2,5,10,4,7,11,6,9,8,3)(12,34,31,42,41,46,15,19,56,25,24,51,26,30,33,57,16,45,47,39,54,35,55,49,48,44,38,37,36,32,29,28,27,21,58,20,18,17,60,23,43,40,14,13,53,52,59,50,22)(61,62)

Conjugacy classes & character table

178 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 39);