Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #6629390 ⋊ C3 |
1727 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 1727 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,3,4,7)(2,6,5,8)(12,20,13)(15,23,16)(17,22,18)(19,24,21);(2,5)(6,8)(9,22)(10,23)(11,20)(12,18,17,13)(14,24)(15,16,19,21);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,21)(15,17)(16,18)(20,24)(22,23)Polycyclic presentation:
2^3,3,2^6|c^2=ej,f^2=j,g^2=j,b^a=be,f^a=fhi,f^b=fhi,f^d=g,g^a=gh,g^b=gh,g^d=fgj,g^f=gj,h^d=ij,h^f=hj,i^d=hij,i^f=ij,i^g=ij86 conjugacy classes — view the full character table.
| maximal subgroup of | #72246181, #72246847, #72246879, … |
| maximal normal subgroup of | #72246181, #72246847, #72246879, … |
| maximal quotient of | C2×#72245692 |
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, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.5*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);