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C3×A5×C41⋊C5 #49147222

order36900
automorphism?
homologyAbelianization (H1) = C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC3×A5×C41⋊C5
unambiguous semidirect productC41⋊(C3×C5×A5)
unambiguous semidirect product(C3×C41)⋊(C5×A5)
unambiguous semidirect product(C41×A5)⋊(C3×C5)
unambiguous semidirect product(C3×C41×A5)⋊C5

Splittings

FittingC3×C41 ⋊ C5×A5
socleC3×C41×A5 ⋊ C5
derivedC41×A5 ⋊ C3×C5
SylowC41 ⋊ C3×C5×A5

Subgroups

subgroupG/H
center, hypercenterC3A5×C41⋊C5
commutator, hypocenterC41×A5C3×C5
FrattiniC1G
FittingC3×C41C5×A5
socleC3×C41×A5C5
cosocleC41C3×C5×A5
LCSG ▷ C41×A5
UCSC1 ◁ C3

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 49
transitive degreeμt(G) = 615

A minimal faithful permutation representation — 2 generators on 49 points:

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

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36900, 623);