Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting, derived, Sylow | #14554657 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #14554657 | C22 |
| Frattini | C33 | C3⋊D32 |
| socle | C3 | #72277729 |
| LCS | G ▷ #14554657 | |
| UCS | C1 | |
1735 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 1735 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,5)(3,11,12)(6,14,19,25,18,24)(7,20)(8,15)(9,10,22)(16,21,27,26,23,17);(2,5)(3,9)(4,13)(6,16)(7,15)(8,20)(10,12)(11,22)(14,21)(17,25)(18,27)(19,23)(24,26);(1,18,16,4,19,23,13,6,27)(2,24,9,8,25,10,7,14,22)(3,26,5,11,21,15,12,17,20)Polycyclic presentation:
2^2,3^7|d^3=i,c^a=c^2,c^b=c^2,d^a=d^2i^2,e^a=e^2,e^d=eh,f^a=f^2,f^b=f^2,f^c=fh^2i,f^d=fg,g^b=g^2,g^c=gi,h^a=hi,h^d=hi^2,i^a=i^263 conjugacy classes — view the full character table.
| maximal quotient of | #72277755 |
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> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.9^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8^2*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);