Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| Fitting | C9×#493 ⋊ C2 |
| Sylow | C9 ⋊ #1712 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #63502 |
| commutator, cosocle | C42×C9 | C23 |
| Frattini | C3×C42 | C22×D3 |
| Fitting | C9×#493 | C2 |
| socle | C22×C3 | #940 |
| hypercenter | #493 | D9 |
| hypocenter | C9 | #1712 |
| LCS | G ▷ C42×C9 ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
420 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^7,3^2|b^2=df,c^2=eg,d^2=f,e^2=g,h^3=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^c=h^2i^2,i^c=i^2126 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#1712, #72244462, #72244482, … |
| maximal normal subgroup of | D9×#1712, #72244462, #72244482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 16595);