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Cayley diagram

D5×2−1+4 #66286

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

Aliases

direct productD5×2−1+4

Splittings

FittingC5×2−1+4 ⋊ C2
SylowC5 ⋊ C2×2−1+4

Subgroups

subgroupG/H
center, FrattiniC2C24×D5
commutator, socle, cosocleC2×C5C25
FittingC5×2−1+4C2
hypercenter2−1+4D5
hypocenterC5C2×2−1+4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ 2−1+4

794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 5 generators on 21 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10605019, D5×#454, #10612016, …
maximal normal subgroup of#10605019, D5×#454, #10612016, …
maximal quotient of#10617774, #10617776, #10617777, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1624);