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Dic16302 #72301520

order65208
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic16302

Splittings

HallC3×C11×C13×C19 ⋊ Q8
HallC11×C13×C19 ⋊ Dic6
HallC3×C13×C19 ⋊ Dic22
HallC3×C11×C19 ⋊ Dic26
HallC3×C11×C13 ⋊ Dic38
HallC13×C19 ⋊ Dic66
HallC11×C19 ⋊ Dic78
HallC11×C13 ⋊ Dic114
HallC3×C19 ⋊ Dic286
HallC3×C13 ⋊ Dic418
HallC3×C11 ⋊ Dic494
SylowC19 ⋊ Dic858
SylowC13 ⋊ Dic1254
SylowC11 ⋊ Dic1482
SylowC3 ⋊ Dic5434

Subgroups

subgroupG/H
center, FrattiniC2C2×D8151
commutator, socle, prime radical, cosocleC2×C3×C11×C13×C19C22
FittingC3×C4×C11×C13×C19C2
hypercenterC4D8151
hypocenterC3×C11×C13×C19Q8
LCSG ▷ C2×C3×C11×C13×C19 ▷ C3×C11×C13×C19
UCSC1 ◁ C2 ◁ C4

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 65208

A minimal faithful permutation representation — 2 generators on 54 points:

(1,2,3,4)(5,8,7,6)(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)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54);(1,5,3,7)(2,6,4,8)(10,11)(13,22)(14,21)(15,20)(16,19)(17,18)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)

Polycyclic presentation:

2^3,3,11,13,19|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^10,f^a=f^12,g^a=g^18

Conjugacy classes & character table

16305 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 11, 13, 19]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^10);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);