Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×S4 |
| commutator, cosocle | C2×C3×A4 | C24 |
| Fitting | C22×C3×D4 | C2×D3 |
| socle | C23×C3 | C23×D3 |
| hypercenter | D4 | D3×S4 |
| hypocenter | C3×A4 | 2−1+4 |
| LCS | G ▷ C2×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C2 ◁ D4 | |
1584 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 1584 conjugacy classes of subgroups — too many to draw. Only its 131 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:
2^5,3^2,2^2|b^2=e,c^2=e,d^a=de,g^a=g^2,c^b=ce,f^b=f^2,h^b=hi,d^c=de,h^f=i,i^f=hi69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157558);