Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #8541 ⋊ D3 |
| derived | C22⋊(C22⋊A4) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C22⋊(C22⋊A4) | C2×C4 |
| Frattini | C25 | C2×S4 |
| Fitting | #8541 | D3 |
| socle | C22 | #1020⋊C4 |
| prime radical | #87588⋊C2 | C2 |
| cosocle | #85645 | C22 |
| LCS | G ▷ C22⋊(C22⋊A4) | |
| UCS | C1 | |
778 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 778 conjugacy classes of subgroups — too many to draw. Only its 17 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|a^2=c,d^b=d^2,e^a=eg,e^b=f,e^c=eij,e^d=f,f^a=fh,f^b=e,f^c=fi,f^d=ef,g^a=gij,g^b=h,g^d=h,h^a=hi,h^b=g,h^d=gh,i^b=ij,i^d=j,j^d=ij33 conjugacy classes — view the full character table.
| maximal subgroup of | #72215886, C2×#72215879, C3×#72215879 |
| maximal normal subgroup of | #72215886, C2×#72215879, C3×#72215879 |
| maximal quotient of | #72218429, C2×#72215879, #72248138, … |
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, 1, F.3);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.5);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);