Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting, Sylow | #5021138 ⋊ C3 |
| derived | C22×H(3,𝔽4) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #85645 |
| commutator, cosocle, hypocenter | C22×H(3,𝔽4) | C2×C3 |
| Frattini, socle | C24 | C2×C22⋊A4 |
| Fitting | #5021138 | C3 |
| LCS | G ▷ C22×H(3,𝔽4) | |
| UCS | C1 ◁ C22 | |
4417 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 4417 conjugacy classes of subgroups — too many to draw. Only its 39 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^a=cg,c^b=cd,d^a=dh,d^b=c,e^b=ef,e^c=ei,e^d=eij,f^b=e,f^c=fj,f^d=fi,g^b=gh,h^b=g48 conjugacy classes — view the full character table.
| maximal subgroup of | #72209838 |
| maximal normal subgroup of | #72209838 |
| maximal quotient of | #72227705, #72227732, #72227791, … |
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> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.3);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.5);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);