Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Sylow | C32 ⋊ #2401 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×C3⋊D3 |
| commutator, socle, cosocle | C23×C32 | C24 |
| Fitting | C32×#396 | C2 |
| prime radical | C24×C32 | C23 |
| hypercenter | #396 | C3⋊D3 |
| hypocenter | C32 | #2401 |
| LCS | G ▷ C23×C32 ▷ C32 | |
| UCS | C1 ◁ C23 ◁ #396 | |
1296 conjugacy classes of subgroups; only the 295 normal subgroups are shown.
This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 295 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^7,3^2|a^2=fg,b^2=f,d^2=e,b^a=be,c^a=cf,h^a=h^2,i^a=i^2,c^b=cg,h^b=h^2,i^b=i^2,h^d=h^2,i^d=i^2132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 42903);