Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×GL(2,𝔽3) (order 192) · Inn(G) = C1 · Out(G) = C22×GL(2,𝔽3) |
| holomorph | G ⋊ Aut(G) = AGL(1,ℤ/8)×AGL(2,𝔽3) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C3 |
| abelian | C32×C8 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C4 | C2×C32 |
| socle, prime radical | C2×C32 | C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2,3,4,5,6,7,8)(9,10,11);(12,13,14)72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 14);