Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C24 ⋊ #6900 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C3×A42 |
| commutator, socle, cosocle | C24×C32 | C33 |
| Fitting, prime radical | C24×C33 | C32 |
| hypercenter | C33 | A42 |
| hypocenter | C24 | #6900 |
| LCS | G ▷ C24×C32 ▷ C24 | |
| UCS | C1 ◁ C32 ◁ C33 | |
648 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
3^5,2^4|b^3=d^2,c^3=e^2,b^a=bd,c^a=ce,f^b=g,f^c=fg,g^b=fg,g^c=f,h^b=fh,h^c=i,i^b=fgi,i^c=hi144 conjugacy classes — view the full character table.
| maximal subgroup of | #72220160, #72220173, C2×#72220143, … |
| maximal normal subgroup of | #72220160, #72220173, C2×#72220143, … |
| maximal quotient of | C2×#72220143, #72245041 |
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");;
gap> coll := SingleCollector(F, [3, 3, 3, 3, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4^2);;
gap> SetPower(coll, 3, F.5^2);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 6, 2, F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 3, F.6);;
gap> SetConjugate(coll, 8, 2, F.6*F.8);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 9, 2, F.6*F.7*F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);