Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| Fitting | C3×#15263 ⋊ C2 |
| Sylow | C3 ⋊ #772778 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C24×D3 |
| commutator, socle, cosocle | C24×C3 | C25 |
| Fitting | C3×#15263 | C2 |
| hypercenter | #15263 | D3 |
| hypocenter | C3 | #772778 |
| LCS | G ▷ C24×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ #15263 | |
At least 6404 conjugacy classes of subgroups; only the 1075 normal subgroups are shown.
Polycyclic presentation:
2^9,3|a^2=i,b^2=h,d^2=ghi,e^2=f,b^a=bh,c^a=ci,d^a=dh,d^b=dg,d^c=df,e^b=egh,e^c=ef,e^d=efghi,j^e=j^2156 conjugacy classes — view the full character table.
| maximal subgroup of | #72255231, D3×#772778, #72255559, … |
| maximal normal subgroup of | #72255231, D3×#772778, #72255559, … |
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, 1, F.9);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 4, F.7*F.8*F.9);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 10, 5, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);