Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C3×#35761 ⋊ C2 |
| Sylow | C3 ⋊ #7695992 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C25×D3 |
| commutator, socle, cosocle | C23×C3 | C26 |
| Fitting | C3×#35761 | C2 |
| hypercenter | #35761 | D3 |
| hypocenter | C3 | #7695992 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #35761 | |
At least 7558 conjugacy classes of subgroups; only the 4029 normal subgroups are shown.
Polycyclic presentation:
2^9,3|b^2=h,b^a=bi,c^a=ch,c^b=cg,e^a=ei,e^b=eh,f^d=fgh,j^e=j^2171 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#7695992 |
| maximal normal subgroup of | D3×#7695992 |
| maximal quotient of | #72271229, #72271317 |
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", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 2, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.8);;
gap> SetConjugate(coll, 10, 5, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);