Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting, derived, Sylow | #475913 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #5464 |
| commutator, Fitting, cosocle, hypocenter | #475913 | C3 |
| Frattini | C23×C4 | C22⋊A4 |
| socle | C22 | #2679⋊C3 |
| LCS | G ▷ #475913 | |
| UCS | C1 ◁ C2×C4 | |
378 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 21 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)(10,11,17)(12,27,13)(18,44,40)(19,29,66)(20,24,42)(21,26,43)(22,52,51)(23,56,49)(25,41,38)(28,62,48)(30,63,35)(31,64,36)(32,34,37)(33,65,61)(39,45,60)(46,68,70)(47,58,57)(50,55,71)(53,67,59)(54,72,69);(2,6)(3,8)(9,34,16,27)(10,45)(11,44)(12,13,32,37)(14,64,15,63)(17,66,43,52)(18,19)(20,67)(21,49)(22,60)(23,29)(24,68)(25,62)(26,58)(28,69)(30,35,31,36)(33,55)(38,61)(39,57,56,40)(41,72,42,71)(46,50)(47,51)(48,65,70,53)(54,59);(1,5,4,7)(2,8,6,3)(9,17)(10,11)(12,65)(13,48)(14,41)(15,42)(16,43)(18,55)(19,33)(20,24)(21,26)(22,46)(23,69)(25,38)(27,66)(28,29)(30,56)(31,39)(32,53)(34,52)(35,57)(36,40)(37,70)(44,45)(47,54)(49,58)(50,60)(51,59)(61,62)(63,72)(64,71)(67,68)Polycyclic presentation:
3,2^9|b^2=f,c^2=f,d^2=h,e^2=gij,g^2=j,i^2=j,b^a=bc,c^a=b,c^b=cf,d^a=ehj,d^b=dh,d^c=dfhi,e^a=dej,e^b=efij,e^c=efgj,e^d=ej,g^a=hi,g^c=gj,h^a=gij,h^c=hj,i^a=gj,i^b=ij40 conjugacy classes — view the full character table.
| maximal subgroup of | #72268621 |
| maximal normal subgroup of | #72268621 |
| maximal quotient of | #72268642 |
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.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.7*F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.2);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.4*F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 7, 1, F.8*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);