Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#63419 |
| commutator, cosocle | C2×C42⋊C9 | C22 |
| Frattini, socle | C23×C3 | C2×S4 |
| Fitting | C22×C3×C42 | D3 |
| hypercenter | C22 | #63419 |
| hypocenter | C42⋊C9 | D4 |
| LCS | G ▷ C2×C42⋊C9 ▷ C42⋊C9 | |
| UCS | C1 ◁ C2 ◁ C22 | |
310 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^2,2^4|d^3=e,f^2=h,g^2=i,b^a=bc,d^a=d^2e^2,e^a=e^2,f^a=fgi,g^a=gi,h^a=hi,f^d=gh,g^d=fg,h^d=i,i^d=hi60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154014);