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Cayley diagram

C5×Dic11 #6349

order220
isoclinismisoclinic class of D11
automorphismAut(G) = C2×C4×AGL(1,𝔽11) (order 880) · Inn(G) = D11 · Out(G) = C2×C4×C5
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×Dic11

Splittings

derived, SylowC11 ⋊ C4×C5
HallC5×C11 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2×C5D11
commutator, hypocenterC11C4×C5
FrattiniC2C5×D11
Fitting, socle, prime radicalC2×C5×C11C2
cosocleC2×C11C2×C5
LCSG ▷ C11
UCSC1 ◁ C2×C5

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 220

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×Dic11, #89166, #89169, …
maximal normal subgroup ofD5×Dic11, #89166, #89169, …
maximal quotient ofC5×C11⋊C8, C2×C5×Dic11, C3×C5×Dic11, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(220, 4);