Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C32 |
No aliases for this group.
| Fitting, Sylow | #14554857 ⋊ C22 |
| derived | C3×#6908 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×#6908 | C22×C3 |
| Frattini | C33 | C3×D3×C3⋊D3 |
| Fitting | #14554857 | C22 |
| socle | C3 | #72214370 |
| LCS | G ▷ C3×#6908 | |
| UCS | C1 | |
1102 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(1,9,43)(2,21,12)(3,27,7)(4,59,72,15,31,81)(5,25,73,6,28,65)(8,51,34,11,44,42)(10,70,41,30,47,64)(13,54,23,20,52,24)(14,75,49,37,55,48)(16,77,22,39,60,46)(17,29,79,19,56,66)(18,57,78,40,53,80)(26,68,35,32,45,62)(33,69,61,58,67,63)(36,76,50,38,74,71);(1,18,15,5,19,17,6,4,40)(2,33,30,11,26,32,8,10,58)(3,41,37,42,36,63,13,35,39,12,16,62,20,61,38,34,14,64)(7,50,46,24,49,48,23,22,71)(9,60,56,54,53,76,25,55,59,27,31,75,28,74,57,52,29,77)(21,72,68,73,67,80,44,66,70,43,47,79,51,78,69,65,45,81);(2,8,11)(3,13,20)(9,28,25)(10,26,33)(12,34,42)(14,36,16)(21,44,51)(27,54,52)(29,31,53)(30,58,32)(35,61,41)(37,39,38)(43,65,73)(45,67,47)(55,60,74)(56,57,59)(62,64,63)(66,78,72)(68,70,69)(75,76,77)(79,81,80)Polycyclic presentation:
2^2,3^7|f^3=i^2,d^b=d^2,d^c=dg,e^a=e^2,e^d=eh,f^b=f^2i,f^e=fh^2,g^b=g^2,g^c=gi,h^a=h^2i,h^b=h^2,h^e=hi,i^b=i^291 conjugacy classes — view the full character table.
| maximal subgroup of | #72243955, #72244026 |
| maximal normal subgroup of | #72243955, #72244026 |
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, 6, F.9^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.8^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);