Finite Group Databasebeta

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C5×#63804 #13087076

order1440
automorphism?
homologyAbelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×#63804

Splittings

HallC32×C5 ⋊ #131
SylowC32 ⋊ C5×#131

Subgroups

subgroupG/H
centerC4×C5C2×D32
commutatorC2×C32C22×C4×C5
FrattiniC4C2×C5×D32
FittingC32×C5×C8C22
socleC2×C32×C5C22×C4
prime radicalC5×#3798C2
cosocleC32×C4C23×C5
hypercenterC5×C8D32
hypocenterC32C5×#131
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C4×C5 ◁ C5×C8

268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1440, 3393);