Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C3×GL(3,𝔽2) (order 1008) · Inn(G) = C1 · Out(G) = C2×C3×GL(3,𝔽2) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23 |
| abelian | C23×C9 |
| multiplicative | (ℤ/152)ˣ, (ℤ/216)ˣ, (ℤ/228)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C23×C3 |
| socle, prime radical | C23×C3 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2)(7,8,9,10,11,12,13,14,15);(3,4);(5,6)72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 18);