Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Fitting, Sylow | #6342426 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×#127⋊C3 |
| commutator, cosocle | C22×C42 | C23×C3 |
| Frattini, socle | C24 | C23×A4 |
| Fitting | #6342426 | C3 |
| hypercenter | C42 | #127⋊C3 |
| hypocenter | C42 | C3×#127 |
| LCS | G ▷ C22×C42 ▷ C42 | |
| UCS | C1 ◁ C22 ◁ C42 | |
4772 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 4772 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^3,3,2^6|b^2=e,c^2=f,g^2=i,h^2=j,b^a=be,c^a=cf,d^b=dj,g^a=gi,g^d=hi,h^a=hj,h^d=gh,i^d=j,j^d=ij88 conjugacy classes — view the full character table.
| maximal subgroup of | #72213802, #72214051, #72214257, … |
| maximal normal subgroup of | #72213802, #72214051, #72214257, … |
| maximal quotient of | #72222670, C2×#72213771, #72268781 |
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, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);