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Cayley diagram

D87 #4306

order174
isoclinismisoclinic class of D87
automorphismAut(G) = D3×AGL(1,𝔽29) (order 4872) · Inn(G) = G · Out(G) = C4×C7
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD87
unambiguous semidirect productC3⋊D29
unambiguous semidirect productC29⋊D3

Splittings

Fitting, socle, derived, HallC3×C29 ⋊ C2
SylowC29 ⋊ D3
SylowC3 ⋊ D29

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C29C2
LCSG ▷ C3×C29
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 87

A minimal faithful permutation representation — 2 generators on 32 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);(2,3)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)

Polycyclic presentation:

2,3,29|b^a=b^2,c^a=c^28

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D29, C2×D87, D261, …
maximal normal subgroup ofD3×D29, C2×D87, C3×D87, …
maximal quotient ofDic87, C2×D87, D261, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(174, 3);