Finite Group DatabasebetaFinite Group Databasebeta| order | 5940 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, derived, Hall | C11×H(3,𝔽3) ⋊ C22×C5 |
| Hall | H(3,𝔽3)×C11⋊C5 ⋊ C22 |
| Hall | #272⋊D11 ⋊ C5 |
| Hall | C11⋊C5 ⋊ ((C3⋊D3)⋊C3)⋊C2 |
| Sylow | H(3,𝔽3) ⋊ C2×AGL(1,𝔽11) |
| Sylow | C11 ⋊ C5×((C3⋊D3)⋊C3)⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C11×H(3,𝔽3) | C22×C5 |
| Frattini | C3 | (C3⋊D3)⋊AGL(1,𝔽11) |
| socle | C3×C11 | C5×D32 |
| LCS | G ▷ C11×H(3,𝔽3) | |
| UCS | C1 | |
156 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 99 |
Polycyclic presentation:
2^2,5,3^2,11,3|d^a=d^2,e^a=e^2,e^b=e^2,e^d=eg,f^b=f^10,f^c=f^4,g^b=g^265 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊C3)⋊C2×AGL(1,𝔽11) |
| maximal normal subgroup of | ((C3⋊D3)⋊C3)⋊C2×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, 3, 3, 11, 3]);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6^10);;
gap> SetConjugate(coll, 6, 3, F.6^4);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);