Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
798 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 798 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,4,3)(5,8,10,24,12,25,11,9)(6,14,16,22,17,15,26,34)(7,19,20,36,21,28,27,23)(13,29,31,30,18,35,32,33);(5,7)(6,13)(8,23)(9,19)(10,27)(11,20)(12,21)(14,33)(15,30)(16,32)(17,18)(22,35)(24,28)(25,36)(26,31)(29,34);(2,4)(5,6,9,15,11,16,25,34,12,17,24,14,10,26,8,22)(7,18,19,33,20,31,36,35,21,13,28,30,27,32,23,29)Polycyclic presentation:
2^7,3,2^2|a^2=d,c^2=e,d^2=f,e^2=g,f^2=g,b^a=bg,c^a=ce,c^b=ceg,e^a=eg,e^b=eg,h^a=h^2,i^a=ij,i^h=j,j^h=ij116 conjugacy classes — view the full character table.
| maximal subgroup of | #72342875, S4×#2171, #72343101 |
| maximal normal subgroup of | #72342875, S4×#2171, #72343101 |
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, 3, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.10);;
gap> SetConjugate(coll, 10, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);