Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Fitting, Sylow | #6342420 ⋊ C3 |
| derived | C44 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C44 | C2×C3 |
| Frattini, socle | C24 | C2×C22⋊A4 |
| Fitting | #6342420 | C3 |
| LCS | G ▷ C44 | |
| UCS | C1 | |
907 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 907 conjugacy classes of subgroups — too many to draw. Only its 30 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=i,f^2=j,c^a=ch,c^b=dg,d^a=dgh,d^b=cd,e^a=eij,e^b=cdfhij,f^a=fi,f^b=defi,g^b=h,h^b=gh,i^b=ghj,j^b=hij56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213872, #72214108, C2×#72213869, … |
| maximal normal subgroup of | #72213872, #72214108, C2×#72213869, … |
| maximal quotient of | #72222710, C2×#72213869, #72237333 |
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.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.3*F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.4*F.5*F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.7*F.8*F.10);;
gap> SetConjugate(coll, 10, 2, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);