Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C8 |
No aliases for this group.
| Fitting | C3×#1712 ⋊ C2 |
| Sylow | C3 ⋊ #19745 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D3×#127 |
| commutator, cosocle | C3×C42 | C24 |
| Frattini | C42 | C23×D3 |
| Fitting | C3×#1712 | C2 |
| socle | C22×C3 | C2×#127 |
| hypercenter | #1712 | D3 |
| hypocenter | C3 | #19745 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #1712 | |
994 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 175 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:
2^8,3|b^2=e,c^2=f,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,i^a=i^2,d^b=dg,d^c=dh,i^d=i^278 conjugacy classes — view the full character table.
| maximal subgroup of | #72214058, D3×#19745, #72215215, … |
| maximal normal subgroup of | D3×#19745, #72215215, C3×#10784429, … |
| maximal quotient of | #72214058, #72225898, #72244472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 158810);