Finite Group DatabasebetaFinite Group Databasebeta| order | 12167 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C232 · Schur multiplier (H2) = C232 |
| unitriangular | UT(3,𝔽23) |
| extraspecial | 23+1+2 |
| Heisenberg | H(3,𝔽23) |
| unambiguous semidirect product | C232⋊C23 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C232 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
51 conjugacy classes of subgroups.
Polycyclic presentation:
23^3|b^a=bc551 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12167, 3);