Finite Group DatabasebetaFinite Group Databasebeta| order | 6600 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, socle, derived | C3×C5×C11 ⋊ C2×C4×C5 |
| Hall | C3×C5×C11⋊C5 ⋊ C2×C4 |
| Hall | C5×C11⋊C5 ⋊ C4×D3 |
| Hall | C3×C11 ⋊ C2×C5×AGL(1,𝔽5) |
| Sylow | C11 ⋊ C5×D3×AGL(1,𝔽5) |
| Sylow | C3 ⋊ AGL(1,𝔽5)×AGL(1,𝔽11) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C11 | C2×C4×C5 |
| prime radical | D5×C3⋊AGL(1,𝔽11) | C2 |
| cosocle | C3×D5×C11 | C22×C5 |
| LCS | G ▷ C3×C5×C11 | |
| UCS | C1 | |
136 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 165 |
Polycyclic presentation:
2^2,5,2,3,5,11|b^2=d,e^a=e^2,f^b=f^3,f^d=f^4,g^a=g^10,g^b=g^10,g^c=g^384 conjugacy classes — view the full character table.
| maximal subgroup of | D3×AGL(1,𝔽5)×AGL(1,𝔽11) |
| maximal normal subgroup of | D3×AGL(1,𝔽5)×AGL(1,𝔽11) |
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, 5, 2, 3, 5, 11]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^3);;
gap> SetConjugate(coll, 6, 4, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7^10);;
gap> SetConjugate(coll, 7, 2, F.7^10);;
gap> SetConjugate(coll, 7, 3, F.7^3);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);