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Cayley diagram

C2×#4147 #66279

order320
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#4147

Splittings

FittingC22×C5×Q8 ⋊ C2
SylowC5 ⋊ C2×2−1+4

Subgroups

subgroupG/H
centerC22C23×D5
commutator, cosocleC2×C5C25
FrattiniC2C24×D5
FittingC22×C5×Q8C2
socleC22×C5C24
hypercenterC22×Q8D5
hypocenterC5C2×2−1+4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×Q8

794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 23 points:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10600485, #10600914, #10601140, …
maximal normal subgroup of#10600485, #10600914, #10601140, …
maximal quotient ofC2×#65834, C2×#65847, C2×#65838, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1617);