Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | #15947 ⋊ D3 |
| derived | C22⋊(C22⋊A4) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊(C22⋊A4) | C23 |
| Frattini, socle | C24 | C22×S4 |
| Fitting | #15947 | D3 |
| LCS | G ▷ C22⋊(C22⋊A4) | |
| UCS | C1 | |
3013 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 3013 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^3,3,2^6|d^c=d^2,e^a=eij,e^b=egh,e^c=fg,e^d=f,f^a=fi,f^b=fg,f^c=egh,f^d=ef,g^c=gh,g^d=h,h^d=gh,i^c=ij,i^d=j,j^d=ij45 conjugacy classes — view the full character table.
| maximal subgroup of | #72209643, #72215886, C2×#72209642, … |
| maximal normal subgroup of | #72215886, C2×#72209642, C3×#72209642 |
| maximal quotient of | #72209764, #72209774, #72217499, … |
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, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 4, 3, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.6*F.7);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);