Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C32×#120 ⋊ C8 |
| Sylow | C32 ⋊ #8965 |
1265 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 1265 conjugacy classes of subgroups — too many to draw. Only its 97 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 — 3 generators on 17 points:
(1,9,5)(2,4,3)(6,8,7)(14,17);(2,4,5,3,6,7,9,8)(10,12)(11,14)(13,16)(15,17);(10,11)(12,14)(13,15)(16,17)Polycyclic presentation:
2^8,3^2|b^2=f,f^2=h,c^a=ce,c^b=cd,d^a=dg,e^b=eg,i^b=j,i^f=ij^2,i^h=i^2,j^b=ij^2,j^f=i^2j^2,j^h=j^278 conjugacy classes — view the full character table.
| maximal subgroup of | #72252956, B(4,𝔽2)×AGL(1,𝔽9), #72253248 |
| maximal normal subgroup of | #72252956, B(4,𝔽2)×AGL(1,𝔽9), #72253248 |
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, 2, F.6);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10^2);;
gap> SetConjugate(coll, 9, 8, F.9^2);;
gap> SetConjugate(coll, 10, 2, F.9*F.10^2);;
gap> SetConjugate(coll, 10, 6, F.9^2*F.10^2);;
gap> SetConjugate(coll, 10, 8, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);