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 | #14555437 ⋊ C22 |
| derived | C3×#737 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×#737 | C22×C32 |
| Frattini | C32 | C32×#1113 |
| Fitting | #14555437 | C22 |
| socle | C3 | C3×(C3⋊#1113)⋊C3 |
| LCS | G ▷ C3×#737 | |
| UCS | C1 | |
2653 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 2653 conjugacy classes of subgroups — too many to draw. Only its 101 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:
(1,3,35,17,14,33)(2,45,74,18,50,69)(4,64,60,39,36,77)(5,12,11)(6,27,73,22,57,81)(7,65,72,9,55,80)(8,67,59,42,70,58)(10,79,28,30,71,44)(13,78,16,63,62,40)(15,38,61,41,37,34)(19,52,66,48,47,51)(20,75,26,24,76,25)(21,32,68,49,43,54)(23,53,31,29,56,46);(1,7,9,5,22,18,17,2,6)(3,51,30,33,32,66,12,31,20,11,24,58,14,59,43,35,10,46)(4,23,26,16,48,44,41,8,21)(13,71,53,60,55,80,36,54,47,34,50,73,38,75,67,62,27,69)(15,25,29,40,49,42,39,28,19)(37,72,57,78,70,79,64,74,45,61,52,76,63,81,65,77,56,68)Polycyclic presentation:
2^2,3^7|e^3=i,e^b=e^2i^2,e^c=ei,e^d=eh,f^a=f^2,g^a=g^2,g^b=g^2,g^f=gi,h^b=h^2,h^d=hi,i^b=i^2155 conjugacy classes — view the full character table.
| maximal subgroup of | #72243955, #72244065 |
| maximal normal subgroup of | #72244065 |
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, 5, F.9);;
gap> SetConjugate(coll, 5, 2, F.5^2*F.9^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);