Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| Fitting, derived, Sylow | #357285 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2 | #11709219 |
| commutator, Fitting, cosocle, hypocenter | #357285 | C3 |
| Frattini | C25 | C22⋊A4 |
| LCS | G ▷ #357285 | |
| UCS | C1 ◁ C2 | |
723 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 723 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
3,2^9|b^2=f,c^2=g,d^2=fgij,e^2=fhi,b^a=cfj,c^a=bcj,c^b=cj,d^a=efgi,d^b=dh,d^c=dhi,e^a=dej,e^b=ehi,e^c=eij,e^d=ej,f^a=g,f^d=fj,g^a=fgj,g^d=gj,g^e=gj,h^a=i,h^b=hj,h^d=hj,h^e=hj,i^a=hi,i^b=ij,i^c=ij,i^e=ij31 conjugacy classes — view the full character table.
| maximal subgroup of | #72210033, #72210042, C2×#72210029, … |
| maximal normal subgroup of | C2×#72210029, #72263265 |
| maximal quotient of | #72238912, #72239178, C2×#72210029, … |
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, [3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.6*F.8*F.9);;
gap> SetConjugate(coll, 2, 1, F.3*F.6*F.10);;
gap> SetConjugate(coll, 3, 1, F.2*F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.4*F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);