Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3×C4 |
No aliases for this group.
| Fitting | C2×C32×C7×D4 ⋊ C2 |
| Hall | C32×C7 ⋊ #127 |
| Sylow | C32 ⋊ #6503 |
| Sylow | C7 ⋊ #64149 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C3⋊D21 |
| commutator, socle, cosocle | C22×C32×C7 | C23 |
| Fitting | C2×C32×C7×D4 | C2 |
| hypercenter | C2×D4 | C3⋊D21 |
| hypocenter | C32×C7 | #127 |
| LCS | G ▷ C22×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
648 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1008 |
Polycyclic presentation:
2^5,3^2,7|a^2=d,c^2=e,b^a=bd,c^b=ce,f^a=f^2,g^a=g^2,h^a=h^6324 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#6503, #127×C3⋊D21, D7×#64149, … |
| maximal normal subgroup of | C3⋊D3×#6503, #127×C3⋊D21, D7×#64149, … |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 3, 7]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);