Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C7×#525 ⋊ C2×C3 |
| Hall | #11743111 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #3490 |
| Sylow | C7 ⋊ C3×#3490 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C25×C3 |
| Fitting | C7×#525 | C2×C3 |
| hypercenter | #525 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#3490 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #525 | |
2656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,7)(4,5,6)(18,23)(20,22);(10,14)(13,15);(8,10)(9,13)(11,14)(12,15)(16,18)(17,20)(19,23)(21,22);(9,12)(10,14)(16,17,19,21)(18,22,23,20);(2,3)(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)Polycyclic presentation:
2^5,3,2^2,7|b^2=g,b^a=bh,c^b=cgh,d^c=dh,e^c=eg,i^a=i^6,i^f=i^2160 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#3490, #72277632, #72277873 |
| maximal normal subgroup of | AGL(1,𝔽7)×#3490, #72277632, #72277873 |
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, 2, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> SetConjugate(coll, 9, 6, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);