Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#3026 ⋊ C2 |
| Sylow | C3 ⋊ #23996 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #82581 |
| commutator, cosocle | C24×C3 | C24 |
| Frattini | C24 | C23×D3 |
| Fitting | C3×#3026 | C2 |
| socle | C2×C3 | #2404 |
| hypercenter | #3026 | D3 |
| hypocenter | C3 | #23996 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ #3026 | |
1396 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^8,3|b^2=e,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,g^b=gh,i^b=i^2,e^c=eh,f^d=fh69 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#23996, #72231044, #72231059, … |
| maximal normal subgroup of | D3×#23996, #72231044, #72231059, … |
| maximal quotient of | #72270009, #72270363, C2×#10827751 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 202132);