Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| Fitting | C2×C3×#381 ⋊ C2 |
| Sylow | C3 ⋊ #14335 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator | C23×C3 | C23×C4 |
| Fitting | C2×C3×#381 | C2 |
| socle, cosocle | C24×C3 | C24 |
| prime radical | C24×D12 | C2 |
| hypercenter | C2×#381 | D3 |
| hypocenter | C3 | #14335 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C2×#381 | |
6494 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|c^2=h,b^a=be,c^a=cf,i^a=i^2,c^b=cg,i^b=i^2,d^c=de120 conjugacy classes — view the full character table.
| maximal subgroup of | #72253612, #72253645, #72253758, … |
| maximal normal subgroup of | #72253612, #72253645, #72253758, … |
| maximal quotient of | #72270468 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 96728);