Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| Fitting | C4×C7×C22⋊C4 ⋊ C2×C3 |
| Hall | #11731032 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #2433 |
| Sylow | C7 ⋊ C3×#2433 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×AGL(1,𝔽7) |
| commutator, socle, cosocle | C23×C7 | C24×C3 |
| Fitting | C4×C7×C22⋊C4 | C2×C3 |
| hypercenter | C4×C22⋊C4 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#2433 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 | |
2696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 19 points:
(6,8)(9,10,11,12)(13,14,19)(15,16,18);(5,6)(7,8);(2,4)(10,12)(14,15)(16,18)(17,19);(1,2,3,4)Polycyclic presentation:
2^4,3,2^3,7|a^2=f,d^2=h,b^a=bf,d^b=dh,d^c=dg,i^b=i^6,i^e=i^4154 conjugacy classes — view the full character table.
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, 3, 2, 2, 2, 7]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);