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AGL(1,𝔽5)×#733 #13108174

order1620
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productAGL(1,𝔽5)×#733

Splittings

Fitting, HallC5×#733 ⋊ C4
SylowC5 ⋊ C4×#733

Subgroups

subgroupG/H
centerC3AGL(1,𝔽5)×H(3,𝔽3)
commutatorC32×C5C32×C4
FrattiniC32C32×AGL(1,𝔽5)
FittingC5×#733C4
socleC3×C5C4×H(3,𝔽3)
prime radical, cosocleC32×D5C2×C32
hypercenter#733AGL(1,𝔽5)
hypocenterC5C4×#733
LCSG ▷ C32×C5 ▷ C3×C5 ▷ C5
UCSC1 ◁ C3 ◁ C32 ◁ #733

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 135

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1620, 80);