Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C25×D4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×#1020 |
| commutator, cosocle | C22×C22⋊A4 | C23 |
| Fitting | C25×D4 | D3 |
| socle | C26 | C22×D3 |
| hypercenter | C2×D4 | #1020 |
| hypocenter | C22⋊A4 | #120 |
| LCS | G ▷ C22×C22⋊A4 ▷ C22⋊A4 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
15908 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 15908 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^5,3,2^4|b^2=d,b^a=be,c^a=ce,c^b=cd,f^b=f^2,g^b=h,g^f=h,h^b=g,h^f=gh,i^b=hj,i^f=j,j^b=gi,j^f=ij76 conjugacy classes — view the full character table.
| maximal subgroup of | #120×#1020, #72220596, #72220729, … |
| maximal normal subgroup of | #120×#1020, #72220596, #72220729, … |
| maximal quotient of | #72222909, #72236671, #72247659, … |
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, 2, 2, 2, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 10, 2, F.7*F.9);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);