Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | #62849 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | #543 | C22×C3 |
| Fitting | #62849 | C3 |
| prime radical | #543⋊C3 | C22 |
| hypercenter | C2×Q8 | C22⋊A4 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
1208 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 1208 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=h,d^2=i,e^2=i,b^a=cdf,c^a=bcef,d^a=dfgi,e^a=efgi,f^a=gi,g^a=fgh,c^b=ch,d^b=di,g^b=gi,f^c=fi,e^d=ei54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085122);