Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C3×#376 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊#857 |
| commutator, cosocle, hypocenter | C3×#95⋊C3 | C22 |
| Frattini | C23 | C2×C3⋊S4 |
| Fitting | C3×#376 | D3 |
| socle | C23×C3 | C2×S4 |
| prime radical | C3⋊#4622 | C2 |
| LCS | G ▷ C3×#95⋊C3 | |
| UCS | C1 ◁ C4 | |
268 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 27 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|b^2=gh,e^2=g,f^2=ghi,c^a=c^2,d^a=d^2,e^a=efhi,f^a=fgh,g^a=gi,h^a=hi,e^c=fhi,f^c=efg,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, 154954);