Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
187 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^3,3,2^5|a^2=c,c^2=i,e^2=i,f^2=i,b^a=bi,d^a=d^2,e^a=efh,f^a=fgi,g^a=gh,e^b=eg,f^b=fh,e^c=eg,f^c=fh,e^d=f,f^d=efi,g^d=h,h^d=gh,f^e=fi30 conjugacy classes — view the full character table.
| maximal subgroup of | #72268901 |
| maximal normal subgroup of | #72268901 |
| maximal quotient of | #72268905 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085758);