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 | #14555438 ⋊ 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 | #14555438 | C22 |
| socle | C3 | C3×(C3⋊#1113)⋊C3 |
| LCS | G ▷ C3×#737 | |
| UCS | C1 | |
2797 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 2797 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:
(1,3,34,5,35,32)(2,45,75,9,64,69)(4,63,59,39,36,77)(6,27,80,7,65,81)(8,53,51,29,71,66)(10,76,28,43,74,26)(11,17,12)(13,78,15,62,61,41)(14,38,60,16,37,33)(18,49,73,22,56,72)(19,52,57,23,47,58)(20,70,44,48,79,25)(21,24,68,50,30,54)(31,42,55,46,40,67);(1,7,9,5,22,18,17,2,6)(3,51,30,34,48,66,35,31,20,11,24,57,12,58,43,32,10,46)(4,23,26,16,40,44,15,8,21)(13,70,53,61,64,80,38,54,47,33,49,73,36,74,67,59,27,69)(14,50,29,39,25,42,41,28,19)(37,81,56,77,52,79,62,75,45,60,71,76,63,72,65,78,55,68)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^2,h^b=h^2,h^d=hi^2,i^b=i^2155 conjugacy classes — view the full character table.
| maximal subgroup of | #72243956, #72244066 |
| maximal normal subgroup of | #72244066 |
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.9^2);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);