Finite Group DatabasebetaFinite Group Databasebeta| order | 3840 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| derived | C5×C22⋊A4 ⋊ C42 |
| Sylow | C5 ⋊ #11711329 |
1584 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 1584 conjugacy classes of subgroups — too many to draw. Only its 45 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)(11,17,14,15)(13,16);(2,3,4,5)(6,7,8,9)(10,11)(12,15,16,14,13,17)Polycyclic presentation:
2^4,3,5,2^4|a^2=c,b^2=cd,e^b=e^2,f^a=f^3,f^b=f^3,f^c=f^4,g^a=gi,g^e=h,h^a=hj,h^b=gh,h^e=gh,i^e=j,j^b=ij,j^e=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72243541, #72243922, AGL(1,𝔽5)×#1020⋊C4 |
| maximal normal subgroup of | #72243541, #72243922, AGL(1,𝔽5)×#1020⋊C4 |
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, 2, 2, 3, 5, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^3);;
gap> SetConjugate(coll, 6, 2, F.6^3);;
gap> SetConjugate(coll, 6, 3, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.7*F.8);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 10, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);