Finite Group DatabasebetaFinite Group Databasebeta| order | 54060 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic13515 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D13515 |
| commutator, hypocenter | C3×C5×C17×C53 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C3×C5×C17×C53 | C2 |
| LCS | G ▷ C3×C5×C17×C53 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 82 |
| transitive degree | μt(G) = 54060 |
A minimal faithful permutation representation — 2 generators on 82 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)(30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82);(1,3,2,4)(6,7)(9,12)(10,11)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(31,82)(32,81)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,65)(49,64)(50,63)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)Polycyclic presentation:
2^2,3,5,17,53|a^2=b,c^a=c^2,d^a=d^4,e^a=e^16,f^a=f^5213518 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, 5, 17, 53]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);