Finite Group DatabasebetaFinite Group Databasebeta| order | 10816 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C132 ⋊ C4×M16 |
365 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 365 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 3 generators on 34 points:
(10,11,12,14)(13,16,17,21)(15,19,20,18)(22,23,29,25)(24,32,34,31)(26,33,30,27);(2,5)(6,8)(9,10,13,16,20,14,18,19,11,15,21,17,12);(1,2,3,6,4,5,7,8)(23,25,24,27)(26,31,28,34)(29,30,33,32)Polycyclic presentation:
2^6,13^2|a^2=d,c^2=e,d^2=f,b^a=bf,g^a=g^8,g^c=g^8,g^d=g^12,g^e=g^12,h^a=g^6h,h^c=h^8,h^d=g^2h,h^e=h^12136 conjugacy classes — view the full character table.
| maximal subgroup of | #72229302 |
| maximal normal subgroup of | #72229302 |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 13, 13]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.7^8);;
gap> SetConjugate(coll, 7, 4, F.7^12);;
gap> SetConjugate(coll, 7, 5, F.7^12);;
gap> SetConjugate(coll, 8, 1, F.7^6*F.8);;
gap> SetConjugate(coll, 8, 3, F.8^8);;
gap> SetConjugate(coll, 8, 4, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 5, F.8^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);