Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
| unambiguous semidirect product | C3⋊#17309 |
| Sylow | C3 ⋊ #17309 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator, socle, prime radical, cosocle | C24×C3 | C24 |
| Fitting | C3×#1446 | C2 |
| hypercenter | #1446 | D3 |
| hypocenter | C3 | #17309 |
| LCS | G ▷ C24×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ #1446 | |
1466 conjugacy classes of subgroups; only the 283 normal subgroups are shown.
This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 283 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 768 |
Polycyclic presentation:
2^8,3|a^2=h,b^2=g,c^2=g,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de,i^d=i^290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 135132);