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AGL(1,𝔽11)×C61⋊C3 #34117757

order20130
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productAGL(1,𝔽11)×C61⋊C3
unambiguous semidirect productD11⋊(C5×C61⋊C3)
unambiguous semidirect product(C11⋊C5)⋊(C2×C61⋊C3)
unambiguous semidirect product(C61×AGL(1,𝔽11))⋊C3

Splittings

Fitting, socle, derived, HallC11×C61 ⋊ C2×C3×C5
HallC11⋊C5×C61⋊C3 ⋊ C2
HallC61×AGL(1,𝔽11) ⋊ C3
HallD11×C61⋊C3 ⋊ C5
HallC11⋊C5×C61 ⋊ C2×C3
HallC11×C61⋊C3 ⋊ C2×C5
HallD11×C61 ⋊ C3×C5
SylowC61 ⋊ C3×AGL(1,𝔽11)
HallC11⋊C5 ⋊ C2×C61⋊C3
HallD11 ⋊ C5×C61⋊C3
SylowC11 ⋊ C2×C5×C61⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC11×C61C2×C3×C5
LCSG ▷ C11×C61
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 72
transitive degreeμt(G) = 671

A minimal faithful permutation representation — 2 generators on 72 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,34)(18,26,31)(21,29,28)(23,32,25)(27,37,44)(30,39,41)(33,42,38)(35,45,36)(40,48,52)(43,50,49)(46,53,47)(51,56,57)(54,58,55)(59,61,60)(63,67,68,66,71,64,65,72,69,70);(1,2,4,8,15,24,35,46,54,59,58,42,20,5,9,16,25,36,37,14,23,33,43,51,53,32,12,19,28,38,47,48,26,7,13,21,30,40,45,22,31,41,49,55,56,39,17,3,6,11,18,27,34,44,52,57,60,61,50,29,10)(62,63,66,71,65,68,72,67,70,69,64)

Conjugacy classes & character table

253 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20130, 86);