Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | C22⋊SL(2,𝔽3) ⋊ C2×C4 |
1001 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 1001 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^3,3,2^5|a^2=c,b^2=i,g^2=i,h^2=i,b^a=bi,d^a=d^2,e^a=ef,g^a=gh,h^a=hi,e^d=f,f^d=ef,g^d=h,h^d=ghi,h^g=hi48 conjugacy classes — view the full character table.
| maximal subgroup of | #72227567, #72227571, #72227579, … |
| maximal normal subgroup of | #72227567, #72227571, #72227579, … |
| maximal quotient of | #72238097, #72267527, C2×#11715653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090034);