Finite Group Databasebeta

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Cayley diagram

C2×#4022 #65812

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

Aliases

direct productC2×#4022

Splittings

FittingC2×C42×C5 ⋊ C2
SylowC5 ⋊ C2×#126

Subgroups

subgroupG/H
centerC23C22×D5
commutator, cosocleC22×C5C24
FrattiniC22C23×D5
FittingC2×C42×C5C2
socleC23×C5C23
prime radicalC23×D5C22
hypercenterC2×C42D5
hypocenterC5C2×#126
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C42

246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 160

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

(1,2)(3,4,6,7,5)(8,10,11,15)(9,13,12,14)(17,21)(20,23);(9,12)(13,14)(16,18,19,22)(17,20,21,23);(4,5)(6,7)(8,9)(10,14)(11,12)(13,15)(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#10601578, #10601622, C4×#4022, …
maximal normal subgroup of#10601578, #10601622, C4×#4022, …
maximal quotient ofC4×#4022, #10605040, #10605045, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1150);