Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
233 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 53 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:
2^6,3,2^2|a^2=d,b^2=c,c^2=e,e^2=f,b^a=bf,g^a=g^2,h^a=hi,h^g=i,i^g=hi112 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#346, #72342730, #72342733, … |
| maximal normal subgroup of | S4×#346, #72342730, #72342733, … |
| maximal quotient of | #72342813 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085868);