Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | (C22⋊C9)⋊C2 |
| commutator, hypocenter | Q8⋊C9 | C2×C8 |
| Frattini | C3×C8 | C2×S4 |
| Fitting | C3×#504 | D3 |
| socle | C2×C3 | C8×S4 |
| prime radical | #63691 | C4 |
| cosocle | #131⋊C9 | C22 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C16 | |
149 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 29 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:
2^4,3^2,2^3|b^2=c,c^2=d,d^2=i,e^3=f,g^2=i,h^2=i,e^a=e^2f^2,f^a=f^2,g^a=gh,h^a=hi,g^e=h,h^e=ghi,h^g=hi120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154005);