Finite Group DatabasebetaFinite Group Databasebeta| order | 7040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C11×#120 ⋊ C2×C5 |
| Hall | #13082245 ⋊ C5 |
| Hall | C11⋊C5 ⋊ #3484 |
| Sylow | C11 ⋊ C5×#3484 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×AGL(1,𝔽11) |
| commutator, socle, cosocle | C22×C11 | C25×C5 |
| Fitting | C2×C11×#120 | C2×C5 |
| hypercenter | C2×#120 | AGL(1,𝔽11) |
| hypocenter | C11 | C5×#3484 |
| LCS | G ▷ C22×C11 ▷ C11 | |
| UCS | C1 ◁ C22 ◁ C2×#120 | |
3800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 5 generators on 23 points:
(7,11)(10,12)(13,14,15,20,23)(16,21,17,22,19);(2,4)(5,7)(6,10)(8,11)(9,12);(5,6,8,9)(7,10,11,12)(14,15)(16,17)(18,19)(20,22)(21,23);(6,9)(10,12);(1,2)(3,4)Polycyclic presentation:
2^5,5,2^2,11|c^2=h,c^b=ch,d^a=dg,e^d=eh,i^c=i^10,i^f=i^9248 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, 2, 5, 2, 2, 11]);;
gap> SetPower(coll, 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.9^10);;
gap> SetConjugate(coll, 9, 6, F.9^9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);