Finite Group DatabasebetaFinite Group Databasebeta| order | 4704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Hall | #13096416 ⋊ C3 |
| Hall | #3862 ⋊ #120 |
| Sylow | C72 ⋊ C3×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×#64422⋊C2 |
| commutator, socle, cosocle | C22×C72 | C23×C3 |
| Fitting | C23×C72 | C22×C3 |
| hypercenter | C23 | #64422⋊C2 |
| hypocenter | C72 | C3×#120 |
| LCS | G ▷ C22×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ C23 | |
587 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 587 conjugacy classes of subgroups — too many to draw. Only its 82 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:
2^3,3,2^2,7^2|a^2=f,b^a=be,c^a=cf,g^a=g^6,g^b=g^6,g^d=g^2,h^a=g^6h,h^b=h^6,h^d=g^6h^486 conjugacy classes — view the full character table.
| maximal subgroup of | #72246136, #72246146, #72246149 |
| maximal normal subgroup of | #72246136, #72246146, #72246149 |
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.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 2, F.7^6);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^6*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 4, F.7^6*F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);