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C3×#23778454 #44477946

order30420
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×#23778454

Splittings

Fitting, socleC3×C5×C132 ⋊ C3×C4
derived, HallC5×C132 ⋊ C32×C4
HallC3×C5×#92788 ⋊ C4
Hall#16810003 ⋊ C32
HallC3×#92788 ⋊ AGL(1,𝔽5)
SylowC132 ⋊ C32×AGL(1,𝔽5)
SylowC5 ⋊ C3×#14523144

Subgroups

subgroupG/H
center, hypercenterC3#23778454
commutator, hypocenterC5×C132C32×C4
FrattiniC1G
Fitting, socleC3×C5×C132C3×C4
prime radicalC3×C5⋊#11759815C2
cosocleC5⋊(C13⋊D13)C2×C32
LCSG ▷ C5×C132
UCSC1 ◁ C3

576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 2535

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 535);