Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #6508692 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | #85645 |
| commutator, cosocle, hypocenter | #62817 | C2×C3 |
| Frattini | C24 | C2×C22⋊A4 |
| Fitting | #6508692 | C3 |
| prime radical | #11710730 | C2 |
| LCS | G ▷ #62817 | |
| UCS | C1 ◁ C22 | |
1794 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 1794 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,7,6,5,3,8)(9,10,13,16)(11,15,17,22,12,23,18,24,19,14,20,21);(11,18)(14,22)(17,19)(21,23);(1,6)(2,3)(4,8)(5,7)(9,17,13,19)(10,21,16,23)(11,20,18,12)(14,24,22,15)Polycyclic presentation:
2,3,2^8|a^2=j,c^2=j,d^2=j,c^a=cg,c^b=cd,d^a=dgh,d^b=c,d^c=dj,e^b=f,e^c=ej,f^b=ef,f^d=fj,g^b=hi,g^d=gi,h^b=ghi,h^c=hi,h^d=hi48 conjugacy classes — view the full character table.
| maximal subgroup of | #72267338 |
| maximal normal subgroup of | #72267338 |
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, [2, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.3);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.8*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);