Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = GL(2,𝔽3)×GL(3,𝔽2) (order 8064) · Inn(G) = C1 · Out(G) = GL(2,𝔽3)×GL(3,𝔽2) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C3 |
| abelian | C23×C32 |
| multiplicative | (ℤ/252)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
96 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)(10,11,12);(5,6)72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 50);