Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C33 |
No aliases for this group.
| Fitting, Sylow | C3×#10624930 ⋊ C22 |
| derived | C32×AΣL(1,𝔽27) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×AΣL(1,𝔽27) | C22×C3 |
| Frattini | C32 | C3×D3×C3⋊(C3⋊D3) |
| Fitting | C3×#10624930 | C22 |
| socle | C32 | C3×((C3⋊D3)⋊C3)⋊D3 |
| LCS | G ▷ C32×AΣL(1,𝔽27) | |
| UCS | C1 | |
5090 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 5090 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 4 generators on 30 points:
(2,3)(4,5,9)(6,21,27,12,11,30)(7,10,19,8,16,17)(13,26,25,22,20,29)(14,28,15,23,18,24);(1,2,3)(5,19)(9,10)(11,25)(14,27)(16,17)(18,20)(21,30)(23,28)(26,29);(4,6,12)(5,11,21)(7,15,22)(8,13,24)(9,18,28)(10,20,23)(14,26,16)(17,27,29)(19,25,30);(5,10,16)(9,17,19)(11,20,14)(18,27,25)(21,23,26)(28,29,30)Polycyclic presentation:
2^2,3^7|d^b=d^2,e^a=e^2,e^d=eh,f^b=f^2,f^c=fi,f^e=fh^2,g^a=gh^2i^2,g^b=g^2,g^e=gi^2,h^a=h^2i,h^b=h^2,h^e=hi,i^b=i^2171 conjugacy classes — view the full character table.
| maximal subgroup of | #72243959, D3×#72221953 |
| maximal normal subgroup of | #72243959, D3×#72221953 |
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, 3, 3, 3, 3, 3, 3]);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.8^2*F.9^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);