Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32 ⋊ #1313 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | Dic32 |
| commutator | C32×C4 | C4×C8 |
| Frattini, hypercenter | C4×C8 | D32 |
| Fitting, cosocle | C32×C4×C8 | C22 |
| socle, prime radical | C22×C32 | #97 |
| hypocenter | C32 | #1313 |
| LCS | G ▷ C32×C4 ▷ C32 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 | |
193 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 193 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^7,3^2|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,b^a=be,d^a=dg,i^a=hi^2,c^b=cg,h^b=h^2,i^b=i^2156 conjugacy classes — view the full character table.
| maximal subgroup of | #72222588, #72222611, #72222621 |
| maximal normal subgroup of | #72222588, #72222611, #72222621 |
| maximal quotient of | #72222493 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 5342);