Finite Group DatabasebetaFinite Group Databasebeta| order | 2496 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C24×C13 ⋊ Dic3 |
| Sylow | C13 ⋊ #5432 |
346 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 156 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,4)(5,7,8)(9,10,13,16,20,14,18,19,11,15,21,17,12);(1,2)(3,4)(6,7)(10,11,12,14)(13,16,17,21)(15,19,20,18)Polycyclic presentation:
2^4,3,13,2^2|a^2=c,b^a=bd,e^a=e^2,f^a=f^8,f^c=f^12,g^a=h,g^e=gh,h^a=g,h^e=g76 conjugacy classes — view the full character table.
| maximal subgroup of | #72242162, #72242175, #261×GL(2,ℤ/4) |
| maximal normal subgroup of | #72242175, #261×GL(2,ℤ/4) |
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, 3, 13, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^8);;
gap> SetConjugate(coll, 6, 3, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 5, F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);