Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C3 |
No aliases for this group.
| Fitting | C32×#127 ⋊ Q8 |
| Sylow | C32 ⋊ #24312 |
1419 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 1419 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 48 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,4,2,8)(3,9,7,5);(11,15)(12,17);(11,14)(12,17)(15,16);(2,4,6,9)(3,7,8,5)(10,11)(12,14)(13,15)(16,17)Polycyclic presentation:
2^8,3^2|a^2=g,d^2=g,f^2=h,b^a=bf,c^a=ch,c^b=ce,d^a=dg,e^a=eh,f^a=fh,f^b=fh,f^c=fh,i^a=i^2j,i^d=j^2,i^g=i^2,j^a=ij,j^d=i,j^g=j^260 conjugacy classes — view the full character table.
| maximal subgroup of | #453×PSU(3,𝔽2) |
| maximal normal subgroup of | #453×PSU(3,𝔽2) |
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, 2, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 1, F.7);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2*F.10);;
gap> SetConjugate(coll, 9, 4, F.10^2);;
gap> SetConjugate(coll, 9, 7, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 4, F.9);;
gap> SetConjugate(coll, 10, 7, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);