Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #4176⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#4176⋊C2 |
| unambiguous semidirect product | #4176⋊C22 |
| unambiguous semidirect product | #4177⋊C22 |
| unambiguous semidirect product | #4184⋊C22 |
| Fitting | C2×#731 ⋊ C22 |
| derived, Sylow | #731 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4176⋊C2 |
| commutator, cosocle, hypocenter | #731 | C23 |
| Frattini | C32 | C2×D32 |
| Fitting | C2×#731 | C22 |
| socle | C2×C3 | ((C3⋊D3)⋊C3)⋊C2 |
| LCS | G ▷ #731 | |
| UCS | C1 ◁ C2 | |
170 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
34 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4176⋊C2, #13047135, #13047136, … |
| maximal normal subgroup of | C4×#4176⋊C2, #13047135, #13047136, … |
| maximal quotient of | #4176⋊Q8, #13047125, #4184⋊Q8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 300);