Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #31076 ⋊ D3 |
| derived | #31076⋊C3 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #85559 |
| commutator, cosocle, hypocenter | #31076⋊C3 | C2 |
| Frattini | C22×C4 | #1020 |
| Fitting | #31076 | D3 |
| socle | C2 | #11714279 |
| LCS | G ▷ #31076⋊C3 | |
| UCS | C1 ◁ C4 | |
538 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 538 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2,3,2^8|c^2=g,d^2=h,e^2=ghi,f^2=hij,i^2=j,b^a=b^2,c^a=dgh,c^b=dg,d^a=cgh,d^b=cd,e^a=cefghi,e^b=fh,e^c=ehij,e^d=ehj,f^a=cdfhj,f^b=efgi,f^c=fghi,f^d=fi,f^e=fgi,g^a=h,g^b=h,g^e=gj,g^f=gj,h^a=g,h^b=gh,h^f=hj,i^a=ghi,i^b=ghi,i^e=ij,i^f=ij33 conjugacy classes — view the full character table.
| maximal quotient of | #72209810, #72227661, #72227664 |
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, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.7*F.8*F.9);;
gap> SetPower(coll, 6, F.8*F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2^2);;
gap> SetConjugate(coll, 3, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.3*F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.6*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.3*F.4*F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.9);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);