Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×#585 ⋊ D3 |
1688 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 1688 conjugacy classes of subgroups — too many to draw. Only its 133 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|c^2=i,e^2=i,g^2=i,h^2=i,d^a=de,e^a=ei,f^a=f^2,g^a=gh,h^a=hi,d^b=di,e^d=ei,g^f=h,h^f=ghi,h^g=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226219, #72228299, #72228335, … |
| maximal normal subgroup of | #72226219, #72228335, #72258723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089103);