Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×S4 |
| commutator | C2×A4 | C22×C8 |
| Fitting | C22×#131 | D3 |
| socle | C23 | C22×C3⋊C8 |
| prime radical | A4×#41 | C4 |
| cosocle | C8×A4 | C23 |
| hypercenter | #131 | S4 |
| hypocenter | A4 | #504 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C8 ◁ #131 | |
440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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=d,d^2=e,e^2=f,g^a=g^2,h^a=hi,c^b=cf,h^g=i,i^g=hi100 conjugacy classes — view the full character table.
| maximal subgroup of | #72342873, S4×#504, #72343077 |
| maximal normal subgroup of | #72342873, S4×#504, #72343077 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088505);