Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C23×D3 (order 48) · Inn(G) = D3 · Out(G) = C23 |
| holomorph | G ⋊ Aut(G) = D3×#10595100 |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×C3⋊C8 |
| unambiguous semidirect product | C3⋊(C3×C8) |
| derived | C3 ⋊ C3×C8 |
| Sylow | C32 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | D3 |
| commutator, hypocenter | C3 | C3×C8 |
| Frattini | C4 | C3×D3 |
| Fitting | C32×C4 | C2 |
| socle, prime radical | C2×C32 | C4 |
| cosocle | C3×C4 | C2×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C4 | |
19 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2,3)(5,6)(7,11,13,8,10,14,9,12);(4,5,6)36 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C3⋊C8, #3710, #3711, … |
| maximal normal subgroup of | D3×C3⋊C8, #3710, #3711, … |
| maximal quotient of | C3×C3⋊C16, C2×C3×C3⋊C8, C3×C9⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 12);