Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×S4 |
| commutator, cosocle | C22×C3×A4 | C23 |
| Fitting | C23×C3×C4 | C2×D3 |
| socle | C24×C3 | C22×D3 |
| hypercenter | C2×C4 | D3×S4 |
| hypocenter | C3×A4 | #121 |
| LCS | G ▷ C22×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
1050 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 69 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,b^a=be,c^a=ce,g^a=g^2,c^b=cd,f^b=f^2,h^b=hi,h^f=i,i^f=hi66 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#85495, #72253012, #72253027 |
| maximal normal subgroup of | D3×#85495, #72253012, #72253027 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 156039);