Finite Group DatabasebetaFinite Group Databasebeta| order | 20700 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #90646⋊D23 |
| derived, Hall | C23×C52⋊C9 ⋊ C22 |
| Hall | C52×C23 ⋊ C2×D9 |
| Hall | C52⋊C9 ⋊ C2×D23 |
| Sylow | C52 ⋊ D9×D23 |
| Sylow | C23 ⋊ #90646⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C23×C52⋊C9 | C22 |
| Frattini | C3 | #3874⋊D23 |
| Fitting, socle | C3×C52×C23 | C2×D3 |
| LCS | G ▷ C23×C52⋊C9 | |
| UCS | C1 | |
100 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 100 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1035 |
Polycyclic presentation:
2^2,3,23,5^2,3|c^3=g^2,c^a=c^2g,d^a=d^22,e^a=ef,f^a=f^4,g^a=g^2,c^b=c^2g,e^b=e^4f^4,g^b=g^2,e^c=f,f^c=e^4f^4288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20700, 109);