Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = GL(2,𝔽3)×B(4,𝔽2) (order 3072) · Inn(G) = C22 · Out(G) = C2×D4×GL(2,𝔽3) |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C32×D4 |
| unambiguous semidirect product | C4⋊(C22×C32) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2)(7,8,9);(3,4,5,6)(10,11,12);(4,6)90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 179);