Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×C9×#99 ⋊ C2 |
| Sylow | C9 ⋊ #3026 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#63444 |
| commutator, cosocle | C23×C9 | C24 |
| Frattini | C23×C3 | C23×D3 |
| Fitting | C2×C9×#99 | C2 |
| socle | C2×C3 | C2×#882 |
| hypercenter | C2×#99 | D9 |
| hypocenter | C9 | #3026 |
| LCS | G ▷ C23×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C2×#99 | |
1200 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^7,3^2|h^3=i,b^a=be,c^a=cf,d^a=dg,h^a=h^2i^2,i^a=i^2,f^b=fg,e^c=eg,h^c=h^2i^2,i^c=i^2117 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#3026 |
| maximal normal subgroup of | D9×#3026 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 99352);