Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×S4 |
| commutator, prime radical | C22×A4 | C22×C4 |
| Fitting | C22×#97 | D3 |
| socle | C24 | C2×C4×D3 |
| cosocle | C2×C4×A4 | C23 |
| hypercenter | #97 | S4 |
| hypocenter | A4 | #446 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C4 ◁ #97 | |
443 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3,2^2|a^2=e,b^2=e,c^2=d,d^2=f,b^a=be,g^a=g^2,h^a=hi,c^b=cf,h^g=i,i^g=hi76 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087254);