Finite Group DatabasebetaFinite Group Databasebeta| order | 4704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Hall | C3×C72 ⋊ C2×#41 |
| Sylow | C72 ⋊ C2×#237 |
| Sylow | C3 ⋊ #13096694 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×D72 |
| commutator, socle, cosocle | C2×C3×C72 | C24 |
| Fitting | C22×C3×C72 | C23 |
| hypercenter | C22 | D3×D72 |
| hypocenter | C3×C72 | C2×#41 |
| LCS | G ▷ C2×C3×C72 ▷ C3×C72 | |
| UCS | C1 ◁ C2 ◁ C22 | |
818 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 818 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^5,3,7^2|a^2=e,b^a=be,c^b=ce,f^c=f^2,g^a=g^6,g^d=g^6,h^a=g^2h,h^d=h^6201 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#13096694, #72269484, #72269515 |
| maximal normal subgroup of | D3×#13096694, #72269484, #72269515 |
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, 2, 3, 7, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 4, F.7^6);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 4, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);