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Cayley diagram

D31 #314

order62
isoclinismisoclinic class of D31
automorphismAut(G) = AGL(1,𝔽31) (order 930) · Inn(G) = G · Out(G) = C3×C5
holomorphG ⋊ Aut(G) = #72263939
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD31
unambiguous semidirect productC31⋊C2

Splittings

Fitting, socle, derived, SylowC31 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC31C2
LCSG ▷ C31
UCSC1

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 31

A minimal faithful permutation representation — 2 generators on 31 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);(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)

Polycyclic presentation:

2,31|b^a=b^30

Conjugacy classes & character table

17 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D31, D31⋊C3, C3×D31, …
maximal normal subgroup ofC2×D31, D31⋊C3, C3×D31, …
maximal quotient ofDic31, C2×D31, C3×D31, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(62, 1);