Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C33 |
No aliases for this group.
| Fitting, Sylow | #14554316 ⋊ C2 |
| derived | C3×#732 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×#732 | C2×C32 |
| Frattini | C33 | C32×C3⋊D3 |
| Fitting | #14554316 | C2 |
| socle | C3 | C3×#92300 |
| LCS | G ▷ C3×#732 | |
| UCS | C1 | |
1624 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 1624 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:
(2,10,12,7,9,24)(3,13)(4,17)(5,6)(8,32,30,21,26,51)(11,61,14,34,31,40)(15,42)(16,45)(18,46)(19,47)(20,39)(22,62,35,36,33,37)(23,54,49,29,27,25)(28,78,41,57,55,68)(38,44)(43,71)(48,73)(50,79,65,59,63,66)(52,80,58,77,56,72)(53,64,70,69,60,81)(67,75)(74,76);(1,3,13)(2,28,72)(4,43,76)(5,20,45)(6,16,39)(7,50,69)(8,53,36)(9,55,77)(10,63,64)(11,59,25)(12,41,80)(14,66,54)(15,67,19)(17,75,73)(18,48,38)(21,58,34)(22,57,49)(23,33,78)(24,65,81)(26,60,62)(27,35,68)(29,31,79)(30,70,37)(32,52,61)(40,51,56)(42,46,74)(44,71,47);(1,12,7,6,9,10,5,2,24)(3,31,35,16,11,22,20,14,33)(4,30,23,18,26,27,19,8,49)(13,62,40,39,36,34,45,37,61)(15,55,58,43,28,52,48,41,56)(17,25,21,47,54,32,46,29,51)(38,79,69,67,59,64,76,66,81)(42,80,68,73,77,57,71,72,78)(44,70,65,74,60,50,75,53,63)Polycyclic presentation:
2,3^7|d^3=h,d^a=d^2h^2,d^b=df,d^c=dh^2,e^a=e^2,e^d=eg,f^a=f^2,f^b=fh,g^a=gh^2,g^e=gh,h^a=h^2118 conjugacy classes — view the full character table.
| maximal subgroup of | #72277684 |
| maximal normal subgroup of | #72277684 |
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, 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.8^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
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);;
gap> SetConjugate(coll, 7, 1, F.7*F.8^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);