Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #14679 ⋊ D3 |
5277 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 5277 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 16 points:
(6,8)(10,16,14)(11,12,15);(1,3,4,2)(11,15)(13,16);(1,2,4,3)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)Polycyclic presentation:
2^5,3,2^4|a^2=d,b^2=d,c^a=ce,f^b=f^2,g^a=gi,g^b=gh,g^f=h,h^a=hj,h^f=gh,i^b=ij,i^f=j,j^f=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #72243922, C4×#11713200, #72246783, … |
| maximal normal subgroup of | C4×#11713200, #72341604, D4×#1020⋊C4 |
| maximal quotient of | #72243922, #72253665, #72342003 |
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, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);