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Cayley diagram

D93 #4417

order186
isoclinismisoclinic class of D93
automorphismAut(G) = D3×AGL(1,𝔽31) (order 5580) · Inn(G) = G · Out(G) = C2×C3×C5
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD93
unambiguous semidirect productC3⋊D31
unambiguous semidirect productC31⋊D3

Splittings

Fitting, socle, derived, HallC3×C31 ⋊ C2
SylowC31 ⋊ D3
SylowC3 ⋊ D31

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C31C2
LCSG ▷ C3×C31
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 93

A minimal faithful permutation representation — 2 generators on 34 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);(2,3)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)

Polycyclic presentation:

2,3,31|b^a=b^2,c^a=c^30

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D31, C2×D93, D279, …
maximal normal subgroup ofD3×D31, C2×D93, C3⋊(D31⋊C3), …
maximal quotient ofDic93, C2×D93, D279, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(186, 5);