Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊SL(2,ℤ/8) |
| derived | C3×#95⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊#857 |
| commutator, hypocenter | C3×#95⋊C3 | C4 |
| Frattini | C24 | C3⋊S4 |
| Fitting | C2×C3×#95 | D3 |
| socle | C24×C3 | S4 |
| prime radical, cosocle | C2×C3×#95⋊C3 | C2 |
| LCS | G ▷ C3×#95⋊C3 | |
| UCS | C1 ◁ C22 | |
250 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 33 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:
2^2,3^2,2^5|a^2=b,e^2=g,f^2=ghi,c^a=c^2,d^a=d^2,e^a=efi,f^a=fghi,g^a=gi,h^a=hi,e^c=fh,f^c=efgh,g^c=ghi,h^c=i,i^c=hi,f^e=fgh54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154143);