Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C3×#13867 ⋊ C2 |
| Sylow | C3 ⋊ #612424 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C24×D3 |
| commutator, socle, cosocle | C24×C3 | C25 |
| Fitting | C3×#13867 | C2 |
| hypercenter | #13867 | D3 |
| hypocenter | C3 | #612424 |
| LCS | G ▷ C24×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ #13867 | |
68052 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^9,3|a^2=h,c^2=fg,d^2=f,c^a=ci,c^b=cf,d^a=di,d^b=df,d^c=dg,e^a=eh,e^c=ef,e^d=ef,j^d=j^2192 conjugacy classes — view the full character table.
| maximal subgroup of | #72248696, #72248746, #72249047, … |
| maximal normal subgroup of | #72248696, #72248746, #72249047, … |
| maximal quotient of | #72255212, #72270243, #72272037, … |
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.8);;
gap> SetPower(coll, 3, F.6*F.7);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);