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 | #476212 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #5464 |
| commutator, Fitting, cosocle, hypocenter | #476212 | C3 |
| Frattini | C23×C4 | C22⋊A4 |
| socle | C22 | #2679⋊C3 |
| LCS | G ▷ #476212 | |
| UCS | C1 ◁ C2×C4 | |
357 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4,3)(6,7,8)(10,15,11)(16,23,22)(17,24,20)(18,19,21);(1,6,5,3)(2,8,7,4)(10,18)(11,20,21,22)(15,24,19,23)(16,17);(1,8,5,4)(2,3,7,6)(9,19,13,15)(10,11,18,21)(12,23,14,24)(16,22,17,20)Polycyclic presentation:
3,2^9|b^2=i,c^2=i,d^2=fhj,e^2=gij,f^2=j,h^2=j,b^a=c,c^a=bci,c^b=ci,d^a=de,d^b=dh,d^c=df,e^a=dg,e^b=efg,e^c=egij,e^d=eij,f^a=hij,f^c=fj,g^a=fhi,g^b=gj,h^a=fgij,h^b=hj40 conjugacy classes — view the full character table.
| maximal subgroup of | #72227428 |
| maximal normal subgroup of | #72227428 |
| 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.9);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.6*F.8*F.10);;
gap> SetPower(coll, 5, F.7*F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);