Finite Group DatabasebetaFinite Group Databasebeta| order | 5832 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C3×#6900 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72214284 |
| commutator, cosocle | C2×C3×C92 | C22×C3 |
| Frattini | C2×C32 | C3×D3×C3⋊D3 |
| Fitting | C2×C3×#6900 | C22 |
| socle | C2×C33 | C3×D32 |
| hypocenter | C3×C92 | C3×D4 |
| LCS | G ▷ C2×C3×C92 ▷ C3×C92 | |
| UCS | C1 ◁ C2 | |
1197 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 1197 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 25 points:
(2,4)(6,7)(9,13,14,11,12,16)(10,15)(17,22,20,24,21,18)(19,25);(1,2,3,4)(5,6,7)(18,19)(20,21)(22,23)(24,25);(8,12,16,10,9,11,15,14,13)Polycyclic presentation:
2^2,3,2,3^5|a^2=d,e^3=h,f^3=i,b^a=bd,e^a=e^2h^2,e^b=e^2h^2,e^c=eh^2,f^a=e^2f,f^b=f^2i^2,f^c=fi^2,g^b=g^2,h^a=h^2,h^b=h^2,i^a=h^2i,i^b=i^2147 conjugacy classes — view the full character table.
| maximal subgroup of | #72259767, #72259878, D3×#14516829 |
| maximal normal subgroup of | #72259767, #72259878, D3×#14516829 |
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, 2, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8^2);;
gap> SetConjugate(coll, 5, 2, F.5^2*F.8^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.9^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.9^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);