Finite Group DatabasebetaFinite Group Databasebeta| order | 3840 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C5×#379 ⋊ C3×C4 |
| Hall | #13028920 ⋊ C3 |
| Sylow | C5 ⋊ #14693⋊C3 |
2348 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 2348 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,4,3)(7,9)(12,16,13)(14,17,15);(2,3,4,5)(6,7)(8,9)(10,17,16,11)(12,15,13,14)Polycyclic presentation:
2^2,3,2^6,5|a^2=d,b^2=d,b^a=be,f^a=fh,f^c=g,g^a=gi,g^c=fg,h^c=i,i^c=hi,j^a=j^3,j^b=j^3,j^d=j^476 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×AΣL(1,𝔽64), #72243722, #72243964 |
| maximal normal subgroup of | AGL(1,𝔽5)×AΣL(1,𝔽64), #72243722, #72243964 |
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", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^3);;
gap> SetConjugate(coll, 10, 2, F.10^3);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);