Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×D4×GL(2,𝔽3) (order 768) · Inn(G) = D4 · Out(G) = C2×GL(2,𝔽3) |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C3 |
| direct product | C32×SD16 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C32 |
| prime radical | C32×D4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C32×C4 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2,3)(7,8,9,10,11,12,13,14);(4,5,6)(8,10)(9,13)(12,14)63 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 107);