Finite Group DatabasebetaFinite Group Databasebeta| order | 53196 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic13299 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D13299 |
| commutator, hypocenter | C3×C11×C13×C31 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C3×C11×C13×C31 | C2 |
| LCS | G ▷ C3×C11×C13×C31 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 62 |
| transitive degree | μt(G) = 53196 |
A minimal faithful permutation representation — 2 generators on 62 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)(32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62);(1,3,2,4)(6,7)(9,18)(10,17)(11,16)(12,15)(13,14)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)Polycyclic presentation:
2^2,3,11,13,31|a^2=b,c^a=c^2,d^a=d^10,e^a=e^12,f^a=f^3013302 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 3, 11, 13, 31]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^30);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);