Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×S4 |
| commutator, cosocle | C23×A4 | C23 |
| Fitting | C23×C22⋊C4 | D3 |
| socle | C25 | C22×D3 |
| prime radical | C24×A4 | C22 |
| hypercenter | C2×C22⋊C4 | S4 |
| hypocenter | A4 | #397 |
| LCS | G ▷ C23×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
1117 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 1117 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3,2^2|a^2=f,b^2=e,b^a=be,c^a=cd,g^a=g^2,h^a=hi,c^b=cf,h^g=i,i^g=hi64 conjugacy classes — view the full character table.
| maximal subgroup of | #72271319, #72271321 |
| maximal normal subgroup of | #72271321 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086155);