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C5×#3524⋊C3 #14511359

order1920
automorphism?
homologyAbelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#3524⋊C3
unambiguous semidirect product#3524⋊(C3×C5)
unambiguous semidirect product(C5×#3524)⋊C3

Splittings

Fitting, HallC5×#3524 ⋊ C3
Sylow#3524 ⋊ C3×C5

Subgroups

subgroupG/H
center, socleC22×C5C23×A4
commutator, cosocleC2×Q8C23×C3×C5
FrattiniC22C23×C5×A4
FittingC5×#3524C3
hypercenterC5×#126A4
hypocenterQ8C3×C5×#41
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C22×C5 ◁ C5×#126

510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 45
transitive degreeμt(G) = 320

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(6,8,9,12)(7,11,10,13)(15,21)(18,29,30)(19,31,28)(20,24)(22,23,36,34,35,37)(26,41,42)(27,43,40)(32,33,44,38,39,45);(7,10)(11,13)(14,43)(15,45)(16,30)(17,42)(18,27)(19,40)(20,36)(21,44)(22,33)(23,38)(24,37)(25,31)(26,29)(28,41)(32,35)(34,39);(6,7)(8,11)(9,10)(12,13)(14,15,17,21)(16,24,25,20)(18,22,28,34)(19,23,29,35)(26,38,40,32)(27,39,41,33)(30,36,31,37)(42,45,43,44)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237057);