Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C32 |
No aliases for this group.
| Fitting, Sylow | #14554344 ⋊ C22 |
| derived | C3×AΣL(1,𝔽27) ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×AΣL(1,𝔽27) | C22×C32 |
| Frattini | C33 | C32×D32 |
| Fitting | #14554344 | C22 |
| socle | C3 | C3×(((C3⋊D3)⋊C3)⋊D3)⋊C3 |
| LCS | G ▷ C3×AΣL(1,𝔽27) | |
| UCS | C1 | |
1458 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 1458 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 2 generators on 81 points:
(1,3,12)(2,46,73,7,28,74)(4,68,64,16,39,81)(5,15,43,6,19,35)(8,59,79,20,52,78)(9,56,31,21,54,48)(10,70,27,22,55,45)(11,65,75,24,37,71)(13,77,29,36,61,23)(14,80,18,38,63,42)(17,69,34,41,40,62)(25,66,53,44,67,58)(26,76,51,32,72,50)(30,49,33,47,57,60);(1,2,24)(3,33,32,12,13,58)(4,9,50)(5,8,20)(6,11,7)(10,48,19,31,22,43)(14,61,56,34,37,74)(15,53,36,35,26,60)(16,29,47)(17,25,45)(18,30,23)(21,41,51)(27,44,42)(28,71,39,55,49,63)(38,79,59,62,67,76)(40,72,66,64,52,78)(46,81,57,70,69,75)(54,77,68,73,65,80)Polycyclic presentation:
2^2,3^7|e^b=e^2,e^c=ei^2,e^d=eg,f^a=f^2,f^e=fh,g^b=g^2,g^d=gi^2,h^a=h^2i,h^b=h^2,h^f=hi,i^b=i^2119 conjugacy classes — view the full character table.
| maximal subgroup of | #72243959, #72253859 |
| maximal normal subgroup of | #72253859 |
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, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.9^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 4, 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, 6, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);