Finite Group DatabasebetaFinite Group Databasebeta| order | 3528 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, socle, derived, Hall | C32×C72 ⋊ C23 |
| Sylow | C72 ⋊ C2×D32 |
| Sylow | C32 ⋊ C2×D72 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C72 | C23 |
| LCS | G ▷ C32×C72 | |
| UCS | C1 | |
366 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^3,3^2,7^2|d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^a=f^6,f^c=f^6,g^a=f^2g,g^c=g^6126 conjugacy classes — view the full character table.
| maximal subgroup of | #72213636, D32×D72, C2×#72213616, … |
| maximal normal subgroup of | #72213636, D32×D72, C2×#72213616, … |
| maximal quotient of | #72222195, #72222228, #72222235, … |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 7, 7]);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 3, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 3, F.7^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);