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×#731 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×#731 | C2×C32 |
| Frattini | C33 | C32×C3⋊D3 |
| Fitting | #14554316 | C2 |
| socle | C3 | C3×#92300 |
| LCS | G ▷ C3×#731 | |
| UCS | C1 | |
1612 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 1612 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)(8,31,30,21,26,50)(11,39,35,33,14,62)(15,42)(16,45)(18,46)(19,47)(20,38)(22,36,41,34,32,63)(23,25,52,29,27,53)(28,66,59,56,40,78)(37,44)(43,71)(48,73)(49,64,70,57,60,79)(51,72,69,77,55,81)(54,80,58,67,68,61)(65,75)(74,76);(1,3,13)(2,28,72)(4,43,76)(5,20,45)(6,16,38)(7,49,67)(8,54,34)(9,40,81)(10,60,80)(11,57,25)(12,59,77)(14,64,53)(15,65,19)(17,75,73)(18,48,37)(21,55,33)(22,56,27)(23,41,78)(24,70,61)(26,68,36)(29,35,79)(30,58,63)(31,69,39)(32,66,52)(42,46,74)(44,71,47)(50,51,62);(1,12,7,5,9,10,6,2,24)(3,11,32,20,35,41,16,14,22)(4,30,23,19,26,27,18,8,52)(13,34,39,45,63,62,38,36,33)(15,28,55,48,59,69,43,40,51)(17,53,21,46,25,31,47,29,50)(37,57,67,76,79,80,65,64,61)(42,77,66,71,81,78,73,72,56)(44,58,60,75,68,70,74,54,49)Polycyclic presentation:
2,3^7|d^3=h^2,d^a=d^2h,d^b=df,d^c=dh,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 | #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.8^2);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.8);;
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, 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);