Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #4174 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C4×#4174 |
| Fitting | C4×AΣL(1,𝔽27) ⋊ C2 |
| derived | H(3,𝔽3) ⋊ C2×C3×C4 |
| Sylow | AΣL(1,𝔽27) ⋊ C2×C4 |
134 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
88 conjugacy classes — view the full character table.
| maximal subgroup of | #4174⋊C8, #13046411, C8×#4174, … |
| maximal normal subgroup of | #4174⋊C8, #13046411, C8×#4174, … |
| maximal quotient of | C8×#4174, #13046557, C4×#66321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 128);