Finite Group Databasebeta

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Cayley diagram

C2×#4016 #65803

order320
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C24

Aliases

direct productC2×#4016

Splittings

SylowC5 ⋊ C2×#125

Subgroups

subgroupG/H
centerC23C22×D5
commutator, cosocleC22×C5C24
FrattiniC22C23×D5
FittingC2×C42×C5C2
socle, prime radicalC23×C5C23
hypercenterC2×C42D5
hypocenterC5C2×#125
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42

226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 320

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10599427, #10599472, #10599483, …
maximal normal subgroup of#10599427, #10599472, #10599483, …
maximal quotient ofC4×#4016, #10605038, #10605041, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1141);