Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C33 |
No aliases for this group.
| Fitting, Sylow | #14554315 ⋊ C2 |
| derived | C3×AΣL(1,𝔽27) ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×AΣL(1,𝔽27) | C2×C32 |
| Frattini | C33 | C32×C3⋊D3 |
| Fitting | #14554315 | C2 |
| socle | C3 | C3×#92300 |
| LCS | G ▷ C3×AΣL(1,𝔽27) | |
| UCS | C1 | |
1690 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 1690 conjugacy classes of subgroups — too many to draw. Only its 77 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:
(1,2)(3,10,12,13,7,20)(4,15,16,24,8,23)(5,27)(9,18)(11,28,32,37,21,46)(14,51,22,66,36,39)(17,44,43,70,25,55)(19,26)(29,41)(30,31,33,50,48,35)(34,74,49,81,61,63)(38,40)(42,57,54,56,52,45)(47,53)(58,69)(59,71)(60,68,65,76,75,79)(62,78)(64,67)(72,77)(73,80);(1,4,15)(2,8,24)(3,34,80)(5,43,54)(6,16,23)(7,49,78)(9,17,52)(10,58,79)(11,60,40)(12,61,71)(13,67,76)(14,64,31)(18,56,55)(19,25,42)(20,72,68)(21,65,41)(22,69,35)(26,45,70)(27,57,44)(28,62,39)(29,63,33)(30,47,74)(32,75,53)(36,77,50)(37,73,66)(38,81,48)(46,59,51);(1,7,10,2,12,13,6,3,20)(4,22,47,8,36,29,16,14,38)(5,21,30,9,32,33,19,11,48)(15,53,39,24,40,66,23,41,51)(17,49,73,25,61,59,43,34,62)(18,50,28,27,31,37,26,35,46)(42,77,68,54,64,79,52,69,76)(44,80,63,70,78,81,55,71,74)(45,75,72,56,60,58,57,65,67)Polycyclic presentation:
2,3^7|d^3=h,c^a=ch,d^a=d^2h^2,d^b=df,d^c=dh,e^a=e^2,e^c=eh,e^d=eg,f^a=f^2,f^b=fh^2,g^d=gh,g^e=gh^2,h^a=h^2118 conjugacy classes — view the full character table.
| maximal subgroup of | #72277778 |
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");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.8^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.8^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);