Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting, Sylow | #354030 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #1437⋊C3 |
| commutator, cosocle, hypocenter | #2273 | C22×C3 |
| Frattini | C25 | C22×A4 |
| Fitting | #354030 | C3 |
| socle | C23 | C2×#120⋊C3 |
| prime radical | #10935⋊C3 | C2 |
| LCS | G ▷ #2273 | |
| UCS | C1 ◁ C4 | |
1336 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 1336 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,5,4,7,10)(3,8,12,9,6,11)(13,14,17,20)(15,19,27,24,16,25,21,26,22,18,23,28);(1,3,4,9)(2,7)(5,12,10,11)(6,8)(13,22,17,27)(14,25,20,28)(15,23)(16,21)(18,26)(19,24)Polycyclic presentation:
2^2,3,2^7|b^2=j,d^2=h,e^2=i,c^b=ch,d^a=dg,d^c=de,e^a=ef,e^c=di,f^c=g,f^d=fj,g^c=fgj,g^e=gj,h^c=hi,i^c=h64 conjugacy classes — view the full character table.
| maximal subgroup of | #72267233 |
| maximal quotient of | C4×#10935⋊C3, #72267501 |
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, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.4*F.9);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);