Finite Group DatabasebetaFinite Group Databasebeta| order | 8172 |
| automorphism | Aut(G) order 5540616 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic2043 |
| unambiguous semidirect product | C9⋊Dic227 |
| unambiguous semidirect product | C227⋊Dic9 |
| derived, Hall | C9×C227 ⋊ C4 |
| Sylow | C227 ⋊ Dic9 |
| Sylow | C9 ⋊ Dic227 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D2043 |
| commutator, hypocenter | C9×C227 | C4 |
| Frattini | C2×C3 | D681 |
| Fitting | C2×C9×C227 | C2 |
| socle | C2×C3×C227 | D3 |
| LCS | G ▷ C9×C227 | |
| UCS | C1 ◁ C2 | |
18 conjugacy classes of subgroups.
Polycyclic presentation:
2,3,227,2,3|a^2=d,b^3=e^2,b^a=b^2e,c^a=c^226,e^a=e^22046 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Dic4086, Dic6129, Dic14301 |
| maximal normal subgroup of | Dic4086 |
| maximal quotient of | Dic6129, Dic14301 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8172, 4);