Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Sylow | C52 ⋊ #3151 |
1810 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 1810 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 22 points:
(2,4)(6,11)(7,12)(9,10)(13,14,16,17,15)(18,19,21,22,20);(6,10)(9,11)(14,15)(16,17)(19,20)(21,22);(1,2)(3,4)(6,9)(7,12)(10,11);(5,6,7,11,8,10,12,9)(19,20)(21,22)Polycyclic presentation:
2^7,5^2|a^2=f,f^2=g,b^a=bf,c^a=cg,d^a=dfg,d^b=de,f^b=fg,f^d=fg,h^a=h^4,h^c=h^4,i^a=h^3i,i^c=i^4152 conjugacy classes — view the full character table.
| maximal subgroup of | #72226589 |
| maximal normal subgroup of | #72226589 |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.8^3*F.9);;
gap> SetConjugate(coll, 9, 3, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);