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Cayley diagram

D220 #89183

order440
isoclinismisoclinic class of Dic110
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD220
unambiguous semidirect productC4⋊D55

Splittings

FittingC4×C5×C11 ⋊ C2
HallC5×C11 ⋊ D4
SylowC11 ⋊ D20
SylowC5 ⋊ D44

Subgroups

subgroupG/H
center, FrattiniC2C2×D55
commutator, socle, cosocleC2×C5×C11C22
FittingC4×C5×C11C2
hypercenterC4D55
hypocenterC5×C11D4
LCSG ▷ C2×C5×C11 ▷ C5×C11
UCSC1 ◁ C2 ◁ C4

32 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);(2,4)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)

Polycyclic presentation:

2^3,5,11|b^2=c,b^a=bc,d^a=d^4,e^a=e^10

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11724872, #11724873, #11724877, …
maximal normal subgroup of#11724872, #11724873, #11724877, …
maximal quotient of#11724924, D440, Dic220, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(440, 36);