Finite Group DatabasebetaFinite Group Databasebeta| order | 7056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Hall | C32×C72 ⋊ C2×D4 |
| Sylow | C72 ⋊ C2×#626 |
| Sylow | C32 ⋊ #11716115 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #72213616 |
| commutator, Fitting, socle, cosocle | C2×C32×C72 | C23 |
| hypocenter | C32×C72 | C2×D4 |
| LCS | G ▷ C2×C32×C72 ▷ C32×C72 | |
| UCS | C1 ◁ C2 | |
784 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^4,3^2,7^2|a^2=d,b^a=bd,e^a=e^2,e^b=e^2,f^a=e^2f,f^b=f^2,g^a=g^6,g^c=g^6,h^a=g^2h,h^c=h^6228 conjugacy classes — view the full character table.
| maximal subgroup of | #72222241, #72222259 |
| maximal normal subgroup of | #72222241, #72222259 |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 3, 7, 7]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 3, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);