Finite Group DatabasebetaFinite Group Databasebeta| order | 8064 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C24×C3 |
No aliases for this group.
| Fitting | C2×C7×H(5,𝔽2) ⋊ C2×C32 |
| Hall | #11744560 ⋊ C32 |
| Sylow | C7 ⋊ C3×#87547 |
4496 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 4496 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 112 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(3,4)(5,7)(9,13)(10,16,11)(14,23,15,20,22,21)(17,18,19);(2,4,6)(3,5,7)(8,9)(10,20)(11,21)(12,13)(14,17)(15,19)(16,22)(18,23);(8,18,12,16)(9,23,13,22)(10,11,17,19)(14,15,20,21)Polycyclic presentation:
2^2,3^2,2^4,7,2|e^2=j,f^2=j,b^a=bj,e^d=f,f^d=efj,f^e=fj,g^d=gh,g^f=gj,h^d=g,h^e=hj,h^f=hj,i^b=i^6,i^c=i^2127 conjugacy classes — view the full character table.
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, 3, 2, 2, 2, 2, 7, 2]);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 4, F.5*F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);