Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| derived | C22⋊SL(2,𝔽3) ⋊ C8 |
314 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 29 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:
2^3,3,2^5|a^2=b,b^2=c,g^2=i,h^2=i,d^a=d^2,e^a=ef,g^a=gh,h^a=hi,e^d=f,f^d=ef,g^d=h,h^d=ghi,h^g=hi60 conjugacy classes — view the full character table.
| maximal subgroup of | #72267838, #72267876 |
| maximal normal subgroup of | #72267838, #72267876 |
| maximal quotient of | #72267841 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088605);