Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
267 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 267 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=gh,e^2=g,f^2=ghi,b^a=bgh,d^a=d^2,e^a=ef,f^a=fgi,g^a=gi,h^a=hi,e^c=eh,f^c=fi,e^d=fh,f^d=efgh,g^d=ghi,h^d=i,i^d=hi,f^e=fgh30 conjugacy classes — view the full character table.
| maximal subgroup of | #72213654, #72231811, #72231922, … |
| maximal normal subgroup of | #72231952, #72232120, #72268830, … |
| maximal quotient of | #72267669, #72268880 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085751);