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C3×C5×C3⋊#10621728 #44478241

order30420
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC3×C5×C3⋊#10621728
unambiguous semidirect productC3⋊(C3×C5×#10621728)

Splittings

Fitting, socle, HallC32×C5×C132 ⋊ C4
derivedC3×C132 ⋊ C3×C4×C5
HallC32×C132 ⋊ C4×C5
HallC5×C132 ⋊ C3×Dic3
SylowC132 ⋊ C3×C5×Dic3
HallC32×C5 ⋊ #10621728
SylowC32 ⋊ C5×#10621728

Subgroups

subgroupG/H
center, hypercenterC3×C5C3⋊#10621728
commutator, hypocenterC3×C132C3×C4×C5
FrattiniC1G
Fitting, socleC32×C5×C132C4
prime radicalC32×C5×C13⋊D13C2
cosocleC3×C13⋊D13C2×C3×C5
LCSG ▷ C3×C132
UCSC1 ◁ C3×C5

448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 5070

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

1980 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 830);