Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #14679 |
| commutator | C27 | C23×C4 |
| Frattini, cosocle | C28 | C24 |
| prime radical | #72225615 | C2 |
| LCS | G ▷ C27 ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C28 ◁ G | |
| LECS | G ▷ C28 ▷ C24 ▷ C1 | |
Only the 1135 normal subgroups are shown; non-normal subgroups are omitted.
Polycyclic presentation:
2^12|b^2=f,b^a=be,c^b=cg,d^b=dh,e^b=ei,e^c=ej,f^a=fi,f^d=fk,g^a=gj,g^d=gl,h^b=hk,h^c=hl202 conjugacy classes — view the full character table.
| maximal subgroup of | #72309406, #72310817, #72310935, … |
| maximal normal subgroup of | #72309406, #72310817, #72310935, … |
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", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);