Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = GL(2,𝔽3)×#5934 (order 9216) · Inn(G) = C1 · Out(G) = GL(2,𝔽3)×#5934 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C3 |
| abelian | C22×C32×C4 |
| multiplicative | (ℤ/273)ˣ, (ℤ/315)ˣ, (ℤ/364)ˣ, (ℤ/468)ˣ, (ℤ/546)ˣ, (ℤ/630)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C23×C32 |
| socle, prime radical | C23×C32 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2,3,4)(9,10,11);(5,6)(12,13,14);(7,8)144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 178);