Finite Group DatabasebetaFinite Group Databasebeta| order | 29988 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #11725077⋊D17 |
| derived, Hall | C72×C17 ⋊ C22×C9 |
| Hall | C17×#89204 ⋊ C22 |
| Hall | (C7⋊D7)⋊D17 ⋊ C9 |
| Hall | #89204 ⋊ C2×D17 |
| Sylow | C72 ⋊ C2×C9×D17 |
| Sylow | C17 ⋊ #11725076⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | #64424⋊D17 |
| commutator, hypocenter | C72×C17 | C22×C9 |
| Fitting, socle, cosocle | C3×C72×C17 | C22×C3 |
| prime radical | C3×(C7⋊D7)⋊D17 | C3 |
| LCS | G ▷ C72×C17 | |
| UCS | C1 ◁ C3 | |
156 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 156 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) = 2142 |
Polycyclic presentation:
2^2,3,7^2,17,3|c^3=g,d^a=d^6,f^a=f^16,d^b=d^6,e^b=e^6,d^c=d^2,e^c=e^2393 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(29988, 174);