Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = D4×GL(2,𝔽3) (order 384) · Inn(G) = C1 · Out(G) = D4×GL(2,𝔽3) |
| holomorph | G ⋊ Aut(G) = B(4,𝔽2)×AGL(2,𝔽3) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2×C3 |
| abelian | C2×C32×C4 |
| multiplicative | (ℤ/91)ˣ, (ℤ/117)ˣ, (ℤ/182)ˣ, (ℤ/234)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C22×C32 |
| socle, prime radical | C22×C32 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3,4)(7,8,9);(5,6)(10,11,12)72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 36);