Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C32 |
No aliases for this group.
| Fitting, derived, Sylow | #14554861 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #14554861 | C22 |
| Frattini | C33 | C3⋊D32 |
| socle | C3 | #72214331 |
| LCS | G ▷ #14554861 | |
| UCS | C1 | |
1546 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 1546 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:
(1,12,17,21,19,38,5,16,14,13,4,47,6,39,41,3,18,49)(2,76,44,50,15,74,20,52,48,23,27,66,8,63,34,53,10,79)(7,58,57,9,24,65,25,33,42,29,43,70,22,31,32,26,51,69)(11,71,37,36,46,45,40,35,73)(28,61,80,60,68,75,54,64,62,56,59,72,55,78,67,30,77,81);(2,7)(3,11)(4,14)(5,6)(8,25)(9,28)(10,32)(12,37)(13,40)(15,42)(16,45)(17,18)(19,41)(20,22)(21,36)(23,30)(24,44)(26,54)(27,57)(29,55)(31,62)(33,67)(34,43)(35,47)(38,46)(39,73)(48,51)(49,71)(50,60)(52,72)(53,56)(58,80)(59,70)(61,66)(63,75)(64,74)(65,77)(68,69)(76,81)(78,79);(1,8,25,5,2,22,6,20,7)(3,29,53,21,26,50,13,9,23)(4,10,51,18,15,24,19,27,43)(11,30,54,40,60,55,36,56,28)(12,31,63,16,58,76,39,33,52)(14,48,32,41,34,57,17,44,42)(35,61,59,46,64,77,71,78,68)(37,72,80,73,81,62,45,75,67)(38,69,66,47,65,79,49,70,74)Polycyclic presentation:
2^2,3^7|f^3=i,c^a=c^2,d^a=d^2,d^b=d^2,d^c=dg,e^a=e^2,e^b=e^2,e^c=eh,f^a=f^2i^2,f^d=fg^2,f^e=fgi^2,g^a=gi^2,g^b=g^2i,g^d=gi,g^e=gi^2,h^b=h^2,h^d=hi^2,i^a=i^270 conjugacy classes — view the full character table.
| maximal subgroup of | #72243793, #72244030 |
| maximal normal subgroup of | #72243793, #72244030 |
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);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
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, 2, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.9^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.7^2);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9^2);;
gap> SetConjugate(coll, 7, 2, F.7^2*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9^2);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
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);