Finite Group DatabasebetaFinite Group Databasebeta| order | 13122 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C35 |
No aliases for this group.
| Fitting, Sylow | #72243615 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C3×#13091503 |
| commutator, cosocle, hypocenter | H(5,𝔽3) | C2×C33 |
| Frattini | C33 | C33×C3⋊D3 |
| Fitting | #72243615 | C2 |
| hypercenter | C32 | #13091503 |
| LCS | G ▷ H(5,𝔽3) | |
| UCS | C1 ◁ C3 ◁ C32 | |
8634 conjugacy classes of subgroups; only the 251 normal subgroups are shown.
This group has 8634 conjugacy classes of subgroups — too many to draw. Only its 251 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:
(1,7,22,6,11,8,5,2,21)(3,51,26,32,9,59,19,33,23,12,30,49,13,25,37,35,14,46)(4,57,29,43,10,69,20,42,24,16,31,56,18,28,44,45,17,48)(15,65,64,62,54,71,40,53,52,39,38,80,36,77,47,67,27,75)(34,78,73,63,50,81,61,66,70,41,74,76,58,55,79,68,60,72);(1,4,16)(2,10,28)(3,36,41)(5,20,45)(6,18,43)(7,17,42)(8,24,48)(9,38,55)(11,31,57)(12,58,62)(13,40,63)(14,54,66)(15,68,19)(21,29,56)(22,44,69)(23,52,72)(25,60,65)(26,64,76)(27,78,30)(32,61,67)(33,74,77)(34,39,35)(37,47,81)(46,73,75)(49,79,80)(50,53,51)(59,70,71);(1,3,12)(2,14,51)(4,15,34)(5,19,35)(6,13,32)(7,30,25)(8,37,49)(9,33,11)(10,38,74)(16,41,39)(17,54,50)(18,36,58)(20,40,61)(21,23,59)(22,26,46)(24,64,73)(27,60,31)(28,66,77)(29,47,79)(42,78,53)(43,63,62)(44,52,70)(45,68,67)(48,81,75)(55,65,57)(56,72,80)(69,76,71)Polycyclic presentation:
2,3^8|d^3=i,d^b=di,e^a=e^2,e^c=eg,e^d=eg,f^a=f^2,f^c=fh,f^d=fg^2i,g^a=g^2i,g^e=gi,g^f=gi^2,h^a=h^2,h^e=hi^2234 conjugacy classes — view the full character table.
| maximal subgroup of | #72243793, #72243840, #72244007 |
| maximal normal subgroup of | #72243793, #72243840, #72244007 |
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, 4, F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.7^2*F.9);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 5, F.8*F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);