Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11711026 |
| commutator, cosocle, hypocenter | C22⋊(#95⋊C3) | C22 |
| Frattini, socle | C25 | C2×S4 |
| Fitting | #7555 | D3 |
| prime radical | #7555⋊C3 | C2 |
| LCS | G ▷ C22⋊(#95⋊C3) | |
| UCS | C1 ◁ C2 | |
1002 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,3,2^7|b^2=j,d^2=h,e^2=i,b^a=bhj,c^a=cij,c^b=c^2,d^a=dg,d^b=dgj,d^c=ehj,e^a=ef,e^b=defghi,e^c=dej,e^d=ej,f^b=fg,f^c=fg,g^c=f,h^c=i,i^b=hij,i^c=hij44 conjugacy classes — view the full character table.
| maximal subgroup of | #72209784, #72255760, #72261431, … |
| maximal normal subgroup of | #72209784, #72255760, #72261431 |
| maximal quotient of | #72209916, #72227714 |
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, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.4*F.5*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 3, F.6);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 9, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);