Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×#568 ⋊ D3 |
750 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 750 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^5,3,2^3|b^2=i,c^2=d,d^2=i,g^2=i,h^2=i,c^a=ce,d^a=di,f^a=f^2,g^a=gh,h^a=hi,e^c=ei,g^f=h,h^f=ghi,h^g=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226316, #72258746, #72258793 |
| maximal normal subgroup of | #72226316, #72258746, #72258793 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087140);