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Cayley diagram

D129 #63110

order258
isoclinismisoclinic class of D129
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD129
unambiguous semidirect productC3⋊D43
unambiguous semidirect productC43⋊D3

Splittings

Fitting, socle, derived, HallC3×C43 ⋊ C2
SylowC43 ⋊ D3
SylowC3 ⋊ D43

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C43C2
LCSG ▷ C3×C43
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 46
transitive degreeμt(G) = 129

A minimal faithful permutation representation — 2 generators on 46 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);(2,3)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)

Polycyclic presentation:

2,3,43|b^a=b^2,c^a=c^42

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D43, C2×D129, D387, …
maximal normal subgroup ofD3×D43, C2×D129, C3⋊(D43⋊C3), …
maximal quotient ofDic129, C2×D129, D387, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(258, 5);