Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting, Sylow | #612326 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #5464 |
| commutator, cosocle, hypocenter | #17072 | C2×C3 |
| Frattini, socle | C24 | C2×C22⋊A4 |
| Fitting | #612326 | C3 |
| prime radical | #17072⋊C3 | C2 |
| LCS | G ▷ #17072 | |
| UCS | C1 ◁ C2×C4 | |
856 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 856 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,3,7)(4,8,5)(9,14,26,16,11,21,13,23,17,10,19,29)(12,22,31,24,18,28,20,30,25,15,27,32);(1,3,6,8)(2,7,4,5)(12,20)(15,24)(18,27)(22,30);(1,5,6,7)(2,8,4,3)(9,12)(10,15)(13,20)(16,24)(17,25)(21,28)(26,31)(29,32)Polycyclic presentation:
2,3,2^8|a^2=i,c^2=j,d^2=j,e^2=j,f^2=j,c^b=cd,d^b=c,d^c=dj,e^b=f,e^c=eg,e^d=eh,f^b=efj,f^c=fghi,f^d=fgi,f^e=fj,g^b=h,h^b=ghi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72266667 |
| maximal normal subgroup of | #72266667 |
| maximal quotient of | #72266643 |
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.9);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
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.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.5*F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);