Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Sylow | C3 ⋊ #13956 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator | C23×C3 | C23×C4 |
| Fitting, prime radical | C23×C3×#41 | C2 |
| socle, cosocle | C24×C3 | C24 |
| hypercenter | C23×#41 | D3 |
| hypocenter | C3 | #13956 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C23×#41 | |
4686 conjugacy classes of subgroups.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|a^2=h,b^2=g,d^2=g,b^a=be,c^a=cf,d^a=de,i^a=i^2,c^b=cg156 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#13956, #72253475, #72253514, … |
| maximal normal subgroup of | D3×#13956, #72253475, #72253514, … |
| maximal quotient of | #72270452 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 92257);