Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | #516323 ⋊ D3 |
| derived | C22⋊#997 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #87588⋊C2 |
| commutator, hypocenter | C22⋊#997 | C2×C4 |
| Frattini, socle | C24 | C2×#1020 |
| Fitting | #516323 | D3 |
| prime radical | C2×#11715850 | C2 |
| cosocle | C2×C22⋊#997 | C22 |
| LCS | G ▷ C22⋊#997 | |
| UCS | C1 ◁ C22 | |
8134 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 8134 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^3,3,2^7|b^2=c,d^a=d^2,e^a=f,e^b=eij,e^d=f,f^a=e,f^b=fi,f^d=ef,g^a=h,g^d=h,g^f=gk,h^a=g,h^d=gh,h^e=hk,i^a=ij,i^d=j,j^d=ij80 conjugacy classes — view the full character table.
| maximal subgroup of | #72252245, C4×#72209653, #72267102, … |
| maximal normal subgroup of | #72252245, C4×#72209653, #72267102, … |
| maximal quotient of | #72246350, #72246783, #72258539, … |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.6);;
gap> SetConjugate(coll, 6, 1, F.5);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);