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Cayley diagram

D5×D11 #6356

order220
isoclinismisoclinic class of D5×D11
automorphismAut(G) = AGL(1,𝔽5)×AGL(1,𝔽11) (order 2200) · Inn(G) = G · Out(G) = C2×C5
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

direct productD5×D11
unambiguous semidirect product(D5×C11)⋊C2
unambiguous semidirect product(C5×D11)⋊C2
unambiguous semidirect productD55⋊C2

Splittings

Fitting, socle, derived, HallC5×C11 ⋊ C22
SylowC11 ⋊ C2×D5
SylowC5 ⋊ C2×D11

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C11C22
LCSG ▷ C5×C11
UCSC1

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 55

A minimal faithful permutation representation — 2 generators on 16 points:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×D11, C2×D5×D11, C3×D5×D11, …
maximal normal subgroup ofAGL(1,𝔽5)×D11, C2×D5×D11, C3×D5×D11, …
maximal quotient ofDic5×D11, D5×Dic11, #89166, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(220, 11);