Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C22×D32 (order 144) · Inn(G) = D32 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | D3×Dic3 |
| unambiguous semidirect product | (C3×D3)⋊C4 |
| derived, Sylow | C32 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32 |
| commutator, hypocenter | C32 | C2×C4 |
| Fitting, socle, cosocle | C2×C32 | C22 |
| prime radical | C2×C3×D3 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2 | |
35 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,2,3,4)(5,6,7)(9,10);(6,7)(8,9,10)18 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic6, #3795, #3796, … |
| maximal normal subgroup of | D3×Dic6, #3795, #3796, … |
| maximal quotient of | D3×C3⋊C8, #3711, Dic32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 20);