Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting, Sylow | #14555221 ⋊ C22 |
| derived | #10624839 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #10624839 | C22×C3 |
| Frattini | C3×C9 | C3×D3×C3⋊D3 |
| Fitting | #14555221 | C22 |
| socle | C3 | #72243785 |
| LCS | G ▷ #10624839 | |
| UCS | C1 | |
759 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 759 conjugacy classes of subgroups — too many to draw. Only its 51 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 — 3 generators on 81 points:
(1,2)(3,80,10,39,14,68)(4,26,51,9,16,58)(5,7)(8,74,25,40,32,46)(11,37,48,12,36,43)(13,44)(15,35,50,49,18,71)(17,76)(19,24)(21,81,27,65,53,67)(22,55,54)(28,60,64,29,59,66)(30,47,69,45,61,52)(31,79)(33,57,70)(34,78)(38,73)(41,42)(56,75)(62,77)(63,72);(1,3,36)(2,21,61,6,8,59)(4,45,46,15,17,77)(5,14,48,19,10,11)(7,53,69,20,25,28)(9,29,68,33,34,72)(12,67,49,56,73,22)(13,79,16,52,74,18)(23,32,64,24,27,30)(26,66,80,57,31,42)(35,41,44,55,43,81)(37,65,71,63,76,54)(38,78,51,47,40,50)(39,70,62,75,58,60);(2,7,24)(6,20,23)(8,25,32)(9,26,58)(12,37,43)(21,53,27)(22,54,55)(28,64,59)(29,66,60)(30,61,69)(31,62,34)(33,57,70)(35,49,71)(39,68,80)(41,56,63)(42,75,72)(44,73,76)(65,81,67)Polycyclic presentation:
2^2,3^7|d^3=hi^2,g^3=i^2,d^a=d^2h^2i,d^b=d^2h^2i,d^c=dh^2i,e^b=e^2,e^d=ei,f^a=fh^2i^2,f^b=f^2,f^d=fg,g^a=g^2i,g^b=gh^2i,g^c=gi,g^f=gh,h^a=h^2,h^b=h^2i^2,h^f=hi,i^a=i^268 conjugacy classes — view the full character table.
| automorphism group of | #6879 |
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.8*F.9^2);;
gap> SetPower(coll, 7, F.9^2);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.8^2*F.9);;
gap> SetConjugate(coll, 4, 2, F.4^2*F.8^2*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8^2*F.9);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8^2*F.9^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8^2*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 2, F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 6, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);