Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
1505 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 1505 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3,2^2|a^2=e,e^2=f,b^a=bf,c^a=cd,g^a=g^2,h^a=hi,h^g=i,i^g=hi88 conjugacy classes — view the full character table.
| maximal subgroup of | #72228080, M16×GL(2,ℤ/4), #72253117, … |
| maximal normal subgroup of | M16×GL(2,ℤ/4), #72253117, #72258898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086970);