Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #14554314 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | #92301 |
| commutator, cosocle, hypocenter | H(5,𝔽3) | C2×C32 |
| Frattini | C33 | C32×C3⋊D3 |
| Fitting | #14554314 | C2 |
| socle | C3 | C3×#92301 |
| LCS | G ▷ H(5,𝔽3) | |
| UCS | C1 ◁ C9 | |
1508 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 1508 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:
(3,10)(4,13)(7,19)(8,22)(11,55,31,28,29,56)(12,27)(14,33)(16,36)(17,39)(20,72,48,45,46,73)(21,44)(23,50)(25,58,54,30,53,57)(26,32)(34,35)(37,78,65,62,63,79)(38,61)(40,67)(42,75,71,47,70,74)(43,49)(51,52)(59,81,77,64,76,80)(60,66)(68,69);(1,23,68,5,52,39,15,8,67)(2,40,13,9,69,33,24,17,34)(3,48,80,12,46,64,32,20,81)(4,50,41,14,51,6,35,22,18)(7,65,30,21,63,58,49,37,57)(10,70,78,27,71,62,26,42,79)(11,75,66,31,74,16,29,47,38)(19,76,28,44,77,56,43,59,55)(25,73,60,53,72,36,54,45,61);(1,3,10)(2,7,19)(4,11,25)(5,12,27)(6,16,36)(8,20,42)(9,21,44)(13,30,28)(14,31,53)(15,32,26)(17,37,59)(18,38,61)(22,47,45)(23,48,70)(24,49,43)(29,54,35)(33,58,56)(34,57,55)(39,64,62)(40,65,76)(41,66,60)(46,71,52)(50,75,73)(51,74,72)(63,77,69)(67,81,79)(68,80,78)Polycyclic presentation:
2,3^7|b^3=h,d^a=d^2,d^c=dg,e^a=e^2,e^c=ef,f^a=f^2,f^d=fh,g^a=g^2,g^e=gh126 conjugacy classes — view the full character table.
| maximal subgroup of | #72277535 |
| maximal normal subgroup of | #72277535 |
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, 2, F.8);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);