Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #14555345 ⋊ C2 |
| derived | C3×C92 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C92 | C2×C32 |
| Frattini, socle | C32 | C32×C3⋊(C3⋊D3) |
| Fitting | #14555345 | C2 |
| LCS | G ▷ C3×C92 | |
| UCS | C1 | |
2644 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 2644 conjugacy classes of subgroups — too many to draw. Only its 231 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,8,9,4,5,6)(3,7)(11,23,25,14,15,16)(12,18)(13,22)(17,34,29,28,26,35)(19,31)(20,21)(24,36,27,32,33,30);(10,19,20,22,21,31,13,12,18)(11,29,30,25,26,36,15,17,32)(14,33,34,23,27,35,16,24,28);(10,15,14,13,25,16,22,11,23)(12,26,24,21,29,27,19,17,33)(18,36,28,31,30,35,20,32,34);(1,5,4,3,9,6,7,2,8)Polycyclic presentation:
2,3^7|d^3=g,e^3=h,f^3=h^2,d^a=d^2g^2,d^c=dg,e^a=e^2h^2,e^c=eh,f^a=f^2h,f^b=fh^2,g^a=g^2,h^a=h^2171 conjugacy classes — view the full character table.
| maximal subgroup of | #72243364 |
| maximal normal subgroup of | #72243364 |
| maximal quotient of | #72243235, #72243764 |
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> SetPower(coll, 6, F.8^2);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);