Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting, Sylow | #153709 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×C42⋊C3 |
| commutator, hypocenter | #95 | C3×C42 |
| Frattini, socle | C25 | C22×A4 |
| Fitting | #153709 | C3 |
| prime radical | C22×#95⋊C3 | C22 |
| cosocle | C22×#95 | C22×C3 |
| hypercenter | #95 | C42⋊C3 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C23 ◁ #95 | |
1775 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 1775 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,3,2^7|a^2=d,b^2=e,f^2=h,g^2=i,b^a=bj,c^a=chj,c^b=cij,f^c=ghj,g^c=fgj,g^f=gj,h^c=i,i^c=hij144 conjugacy classes — view the full character table.
| maximal subgroup of | #72222575, #72222670, #72222705, … |
| maximal normal subgroup of | #72222575, #72222670, #72222705, … |
| maximal quotient of | C2×#72222564 |
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, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);