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C2×C3×C5⋊#11759815 #44478160

order30420
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×C5⋊#11759815
unambiguous semidirect product(C2×C5)⋊(C3×#11759815)

Splittings

Fitting, socleC2×C3×C5×C132 ⋊ C2×C3
derived, HallC5×C132 ⋊ C22×C32
HallC3×C5×#92788 ⋊ C22
HallC2×C5⋊(C13⋊D13) ⋊ C32
HallC3×#92788 ⋊ C2×D5
SylowC132 ⋊ C2×C32×D5
SylowC5 ⋊ C2×C3×#11759815

Subgroups

subgroupG/H
center, hypercenterC2×C3C5⋊#11759815
commutator, cosocle, hypocenterC5×C132C22×C32
FrattiniC1G
Fitting, socleC2×C3×C5×C132C2×C3
LCSG ▷ C5×C132
UCSC1 ◁ C2×C3

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 5070

A minimal faithful permutation representation — 3 generators on 36 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 749);