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Cayley diagram

Dic31 #1243

order124
isoclinismisoclinic class of D31
automorphismAut(G) = C2×AGL(1,𝔽31) (order 1860) · Inn(G) = D31 · Out(G) = C2×C3×C5
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic31
unambiguous semidirect productC31⋊C4

Splittings

derived, SylowC31 ⋊ C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D31
commutator, hypocenterC31C4
Fitting, socle, prime radical, cosocleC2×C31C2
LCSG ▷ C31
UCSC1 ◁ C2

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 124

A minimal faithful permutation representation — 2 generators on 35 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);(1,3,2,4)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic62, C4×D31, C2×Dic31, …
maximal normal subgroup ofDic62, C4×D31, C2×Dic31, …
maximal quotient ofC31⋊C8, C2×Dic31, C3×Dic31, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(124, 1);