Finite Group DatabasebetaFinite Group Databasebeta| order | 7728 |
| automorphism | Aut(G) order 4080384 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D3864 |
| Fitting | C3×C7×C8×C23 ⋊ C2 |
| Hall | C3×C7×C23 ⋊ D8 |
| Hall | C7×C23 ⋊ D24 |
| Hall | C3×C23 ⋊ D56 |
| Sylow | C23 ⋊ D168 |
| Hall | C3×C7 ⋊ D184 |
| Sylow | C7 ⋊ D552 |
| Sylow | C3 ⋊ D1288 |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 3864 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8)(9,10,11)(12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41);(2,8)(3,7)(4,6)(10,11)(13,18)(14,17)(15,16)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)Polycyclic presentation:
2^4,3,7,23|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^6,g^a=g^221935 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D7728, D11592, D19320, … |
| maximal normal subgroup of | D7728 |
| maximal quotient of | D7728, Dic3864, D11592, … |
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, 2, 2, 3, 7, 23]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);