Finite Group DatabasebetaFinite Group Databasebeta| order | 21780 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | #11925890⋊(C3⋊D3) |
| Fitting, socle, Hall | C32×C112 ⋊ C2×D5 |
| derived, Hall | C32×#10597245 ⋊ C22 |
| Hall | #10597245 ⋊ C2×C3⋊D3 |
| Sylow | C112 ⋊ D5×C3⋊D3 |
| Sylow | C32 ⋊ #11925890⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×#10597245 | C22 |
| Fitting, socle | C32×C112 | C2×D5 |
| LCS | G ▷ C32×#10597245 | |
| UCS | C1 | |
204 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,3^2,5,11^2|c^a=c^2,d^a=d^2,e^a=e^4,f^a=f^5g^8,g^a=f^8g^6,e^b=e^4,f^b=f^6g^3,g^b=f^3g^5,f^e=f^9g^3,g^e=g^5153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21780, 496);