Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
No aliases for this group.
| Fitting | C3×#35222 ⋊ C2 |
| Sylow | C3 ⋊ #7630939 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C25×D3 |
| commutator, socle, cosocle | C23×C3 | C26 |
| Fitting | C3×#35222 | C2 |
| hypercenter | #35222 | D3 |
| hypocenter | C3 | #7630939 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #35222 | |
73092 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^9,3|b^2=h,f^2=i,b^a=bh,c^b=cg,d^b=dgh,e^b=ei,f^c=fi,j^d=j^2192 conjugacy classes — view the full character table.
| maximal subgroup of | #72248597, #72248972, D3×#7630939, … |
| maximal normal subgroup of | #72248597, #72248972, D3×#7630939, … |
| maximal quotient of | #72255231, #72271116, #72271125 |
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> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);