Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C7×D42 ⋊ C2×C3 |
| Hall | #11743676 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #3498 |
| Sylow | C7 ⋊ C3×#3498 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C25×C3 |
| Fitting | C7×D42 | C2×C3 |
| hypercenter | D42 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#3498 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ D42 | |
3496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 5 generators on 19 points:
(5,6,11)(7,8,10)(14,18)(17,19);(2,4)(12,14)(13,17)(15,18)(16,19);(6,7)(8,10)(9,11)(12,13,15,16)(14,17,18,19);(2,4)(13,16)(17,19);(1,2)(3,4)(13,16)(17,19)Polycyclic presentation:
2^5,3,2^2,7|c^2=h,b^a=bg,c^a=ch,c^b=ch,d^a=dg,e^d=eh,i^c=i^6,i^f=i^4157 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#3498 |
| maximal normal subgroup of | AGL(1,𝔽7)×#3498 |
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, 2, 2, 2, 3, 2, 2, 7]);;
gap> SetPower(coll, 3, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 9, 3, F.9^6);;
gap> SetConjugate(coll, 9, 6, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);