Finite Group DatabasebetaFinite Group Databasebeta| order | 3520 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C11×C22⋊C4 ⋊ C2×C5 |
| Hall | #10623024 ⋊ C5 |
| Hall | C11⋊C5 ⋊ #534 |
| Sylow | C11 ⋊ C5×#534 |
692 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 692 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 4 generators on 23 points:
(6,10)(7,11)(13,14,15,20,23)(16,21,17,22,19);(2,4)(5,7,8,11)(6,9,10,12)(14,15)(16,17)(18,19)(20,22)(21,23);(7,11)(9,12);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)Polycyclic presentation:
2^4,5,2^2,11|a^2=f,c^2=g,c^a=cf,c^b=cg,d^a=dg,d^c=dg,h^c=h^10,h^e=h^9130 conjugacy classes — view the full character table.
| maximal subgroup of | #72269486, #72269537, #534×AGL(1,𝔽11) |
| maximal normal subgroup of | #72269486, #72269537, #534×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", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 5, 2, 2, 11]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 8, 3, F.8^10);;
gap> SetConjugate(coll, 8, 5, F.8^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);