Finite Group DatabasebetaFinite Group Databasebeta| order | 4704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C72×C22⋊C4 ⋊ C2×C3 |
| Hall | #13096523 ⋊ C3 |
| Hall | #3861 ⋊ #120 |
| Sylow | C72 ⋊ C3×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×#64424 |
| commutator, socle, cosocle | C22×C72 | C23×C3 |
| Fitting | C72×C22⋊C4 | C2×C3 |
| hypercenter | C22⋊C4 | #64424 |
| hypocenter | C72 | C3×#120 |
| LCS | G ▷ C22×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
1300 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 1300 conjugacy classes of subgroups — too many to draw. Only its 151 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^3,3,2^2,7^2|a^2=f,b^a=bf,c^a=ce,g^b=g^6,g^d=g^4,h^b=h^6,h^d=h^4122 conjugacy classes — view the full character table.
| maximal subgroup of | #120×#64424, #72215150, #72246104, … |
| maximal normal subgroup of | #120×#64424, #72215150, #72246104, … |
| maximal quotient of | #72246176, #72246247, #72246254 |
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, 3, 2, 2, 7, 7]);;
gap> SetPower(coll, 1, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 7, 2, F.7^6);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);