Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×S4 |
| commutator | C22×A4 | C22×C4 |
| Fitting | C25×C4 | D3 |
| socle | C25 | C22×D3 |
| prime radical | C24×A4 | C22 |
| cosocle | C23×A4 | C23 |
| hypercenter | C23×C4 | S4 |
| hypocenter | A4 | #380 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ C23×C4 | |
1801 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 1801 conjugacy classes of subgroups — too many to draw. Only its 149 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,h^g=i,i^g=hi88 conjugacy classes — view the full character table.
| maximal subgroup of | #72253055, #72253059, #72267948, … |
| maximal normal subgroup of | #72253059, #72267949, #72268687 |
| maximal quotient of | #72253059, #72253117, #72253127 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086116);