Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| Fitting | C9×#381 ⋊ C2 |
| Sylow | C9 ⋊ #2280 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D9 |
| commutator | C22×C9 | C23×C4 |
| Frattini, socle | C23×C3 | C23×D3 |
| Fitting | C9×#381 | C2 |
| prime radical | C24×D9 | C22 |
| cosocle | C23×C9 | C24 |
| hypercenter | #381 | D9 |
| hypocenter | C9 | #2280 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C23 ◁ #381 | |
1632 conjugacy classes of subgroups; only the 294 normal subgroups are shown.
This group has 1632 conjugacy classes of subgroups — too many to draw. Only its 294 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^7,3^2|a^2=g,d^2=e,h^3=i,b^a=be,c^a=cf,h^c=h^2i^2,i^c=i^2,h^d=h^2i^2,i^d=i^2168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 35049);