Finite Group DatabasebetaFinite Group Databasebeta| order | 13122 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #72265023 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle | C3×#6911 | C2×C32 |
| Frattini | C34 | C32×C3⋊D3 |
| Fitting | #72265023 | C2 |
| socle | C3 | #72277541 |
| hypocenter | C3×#732 | C2×H(3,𝔽3) |
| LCS | G ▷ C3×#6911 ▷ C3×#732 | |
| UCS | C1 | |
5238 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 5238 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(2,7)(3,14)(4,50,20,18,19,51)(5,6)(8,52,26,22,25,31)(9,10)(11,33)(12,27,38,37,30,54)(13,24)(15,42)(16,74,48,45,46,77)(17,49)(21,41)(23,35)(28,72,55,57,43,78)(29,60)(32,61)(34,39)(36,81,44,66,62,73)(40,76,71,47,70,75)(53,68,63,59,64,79)(56,69,67,80,58,65);(1,3,14)(2,28,65)(4,16,40)(5,21,49)(6,17,41)(7,53,66)(8,36,35)(9,55,80)(10,63,81)(11,59,12)(13,43,69)(15,68,38)(18,47,45)(19,46,70)(20,48,71)(22,67,33)(23,57,37)(24,64,73)(25,44,39)(26,62,61)(27,32,78)(29,79,30)(31,56,60)(34,72,54)(42,52,58)(50,76,74)(51,75,77);(1,13,7,6,9,10,5,2,24)(3,29,34,17,11,23,21,15,32)(4,38,22,20,30,31,19,12,52)(8,54,18,25,37,50,26,27,51)(14,61,42,41,35,33,49,39,60)(16,62,64,48,36,53,46,44,63)(28,67,75,43,56,47,55,58,76)(40,80,72,71,65,57,70,69,78)(45,79,73,74,59,66,77,68,81)Polycyclic presentation:
2,3^8|e^3=i,c^b=cd,e^a=e^2i^2,e^b=eg,e^d=ei^2,f^a=f^2,f^e=fh,g^a=g^2,g^c=gi,h^a=hi^2,h^f=hi,i^a=i^2162 conjugacy classes — view the full character table.
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, 3, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.9^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.9^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*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);