Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C26×C7 ⋊ C2×C3 |
| Hall | #11741573 ⋊ C3 |
| Sylow | C7 ⋊ AΣL(1,𝔽64) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#10620796 |
| commutator, cosocle | C25×C7 | C22×C3 |
| Frattini | C23 | C2×A4×D7 |
| Fitting | C26×C7 | C2×C3 |
| socle | C23×C7 | C22×A4 |
| hypercenter | C22 | #10620796 |
| hypocenter | C24×C7 | C3×D4 |
| LCS | G ▷ C25×C7 ▷ C24×C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
1254 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,4)(7,8,11)(9,10,12)(13,14,16,18,19,17,15);(1,2)(3,4)(5,12,11,6)(7,10,8,9)(14,15)(16,17)(18,19)Polycyclic presentation:
2^2,3,2^5,7|b^a=bd,e^a=eg,e^c=ef,f^a=fh,f^c=e,g^c=gh,h^c=g,i^a=i^6124 conjugacy classes — view the full character table.
| maximal subgroup of | #72277787 |
| maximal normal subgroup of | #72277787 |
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, 3, 2, 2, 2, 2, 2, 7]);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.5);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.7);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);