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C32×#4005 #13086257

order1440
automorphism?
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC32×#4005

Splittings

HallC32×C5 ⋊ #101
SylowC5 ⋊ C32×#101

Subgroups

subgroupG/H
centerC2×C32C22⋊AGL(1,𝔽5)
commutatorC22×C5C2×C32×C4
FrattiniC22C2×C32×AGL(1,𝔽5)
FittingC2×C32×C4×C5C4
socleC2×C32×C5C22⋊C4
prime radicalC22×C32×C5C2×C4
cosocleC2×Dic5C22×C32
hypercenterC2×C32×C4AGL(1,𝔽5)
hypocenterC5C32×#101
LCSG ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C32 ◁ C22×C32 ◁ C2×C32×C4

228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 720

A minimal faithful permutation representation — 2 generators on 27 points:

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

Conjugacy classes & character table

171 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 2574);