Finite Group Databasebeta

← Prev (#4352) Next (#4354) →

Cayley diagram

D89 #4353

order178
isoclinismisoclinic class of D89
automorphismAut(G) = AGL(1,𝔽89) (order 7832) · Inn(G) = G · Out(G) = C4×C11
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD89
unambiguous semidirect productC89⋊C2

Splittings

Fitting, socle, derived, SylowC89 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC89C2
LCSG ▷ C89
UCSC1

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 89
transitive degreeμt(G) = 89

A minimal faithful permutation representation — 2 generators on 89 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(2,89)(3,88)(4,87)(5,86)(6,85)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)

Polycyclic presentation:

2,89|b^a=b^88

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#67063, C2×D89, C3×D89, …
maximal normal subgroup of#67063, C2×D89, C3×D89, …
maximal quotient ofDic89, C2×D89, C3×D89, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(178, 1);