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Cayley diagram

Dic59 #6629

order236
isoclinismisoclinic class of D59
automorphismAut(G) = C2×AGL(1,𝔽59) (order 6844) · Inn(G) = D59 · Out(G) = C2×C29
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic59
unambiguous semidirect productC59⋊C4

Splittings

derived, SylowC59 ⋊ C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D59
commutator, hypocenterC59C4
Fitting, socle, prime radical, cosocleC2×C59C2
LCSG ▷ C59
UCSC1 ◁ C2

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 63
transitive degreeμt(G) = 236

A minimal faithful permutation representation — 2 generators on 63 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);(1,3,2,4)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)

Polycyclic presentation:

2^2,59|a^2=b,c^a=c^58

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic118, C4×D59, C2×Dic59, …
maximal normal subgroup ofDic118, C4×D59, C2×Dic59, …
maximal quotient ofC59⋊C8, C2×Dic59, C3×Dic59, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(236, 1);