Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×D3×D4 (order 96) · Inn(G) = D3 · Out(G) = C2×D4 |
| holomorph | G ⋊ Aut(G) = D3×#11865254 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×Dic3 |
| unambiguous semidirect product | (C2×C3)⋊(C3×C4) |
| derived | C3 ⋊ C2×C3×C4 |
| Sylow | C32 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D3 |
| commutator, hypocenter | C3 | C2×C3×C4 |
| Frattini | C2 | C2×C3×D3 |
| Fitting, socle, prime radical | C22×C32 | C2 |
| cosocle | C2×C3 | C22×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C22×C3 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2)(4,5)(6,7,8)(9,10,11,12);(1,2)(3,4,5)36 conjugacy classes — view the full character table.
| maximal subgroup of | Dic32, (C2×D3)⋊Dic3, #3722, … |
| maximal normal subgroup of | Dic32, (C2×D3)⋊Dic3, #3722, … |
| maximal quotient of | C2×C3×C3⋊C8, C3×#208, C3×C4×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 29);