Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | C9×C9⋊H(3,𝔽3) ⋊ C2 |
| derived | C32×C92 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×C92 | C2×C3 |
| Frattini, socle | C33 | C3×C3⋊(C3⋊D3) |
| Fitting | C9×C9⋊H(3,𝔽3) | C2 |
| LCS | G ▷ C32×C92 | |
| UCS | C1 | |
1699 conjugacy classes of subgroups.
| transitive degree | μt(G) = 729 |
Polycyclic presentation:
2,3^7|d^3=g,e^3=h,c^a=c^2,c^b=cf,d^a=d^2g^2,d^b=dg,e^a=e^2h^2,e^b=eg^2,f^a=f^2,g^a=g^2,h^a=h^2234 conjugacy classes — view the full character table.
| maximal subgroup of | #72243764, #72243962, #72254062, … |
| maximal normal subgroup of | #72243764, #72243962, #72254062, … |
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");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);