Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, derived, Sylow | #355895 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C22⋊(C42⋊C3) |
| commutator, Fitting, cosocle, hypocenter | #355895 | C3 |
| Frattini | C23×C4 | C22⋊A4 |
| socle | C22 | C22⋊(#95⋊C3) |
| LCS | G ▷ #355895 | |
| UCS | C1 ◁ C2×C4 | |
884 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 884 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 72 points:
(2,3,7)(5,6,8)(10,17,39)(11,24,23)(12,13,34)(18,21,27)(19,40,66)(20,49,62)(22,28,43)(25,54,45)(26,57,48)(29,58,50)(30,33,38)(31,36,64)(32,37,63)(35,59,53)(41,51,72)(42,68,55)(44,52,71)(46,60,69)(47,61,67)(56,65,70);(2,6)(3,8)(9,30,16,34)(10,45,23,50)(11,49,17,59)(12,38,33,13)(14,63,15,64)(18,67,43,69)(19,35,48,20)(21,70,22,68)(24,66,39,57)(25,40,29,26)(27,71,28,72)(31,37,32,36)(41,65,44,42)(46,52,47,51)(53,58,62,54)(55,61,56,60);(1,5,4,7)(2,8,6,3)(9,43,14,39,16,18,15,24)(10,28,22,11,23,27,21,17)(12,41,31,53,33,44,32,62)(13,42,36,54,38,65,37,58)(19,55,47,29,48,56,46,25)(20,60,52,40,35,61,51,26)(30,69,63,57,34,67,64,66)(45,72,68,49,50,71,70,59)Polycyclic presentation:
3,2^9|b^2=f,c^2=g,d^2=j,e^2=j,f^2=j,g^2=j,h^2=j,i^2=j,b^a=bc,c^a=bgh,c^b=ch,d^a=de,d^c=dhij,e^a=d,e^b=ehij,e^d=ej,f^a=fgh,f^c=fj,g^a=fj,g^b=gj52 conjugacy classes — view the full character table.
| maximal subgroup of | #72267916, #72267958, #72287613 |
| maximal normal subgroup of | #72267916 |
| maximal quotient of | #72268064 |
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", "j");;
gap> coll := SingleCollector(F, [3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.4);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);