Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C2×D3×D4 (order 96) · Inn(G) = C2×D3 · Out(G) = C23 |
| holomorph | G ⋊ Aut(G) = D3×#11864777 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| direct product | C3×Dic6 |
| unambiguous semidirect product | C3⋊(C3×Q8) |
| Sylow | C32 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D3 |
| commutator, cosocle | C2×C3 | C22×C3 |
| Frattini | C2 | C2×C3×D3 |
| Fitting | C32×C4 | C2 |
| socle, prime radical | C2×C32 | C22 |
| hypercenter | C3×C4 | D3 |
| hypocenter | C3 | C3×Q8 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
27 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)(4,5,6)(7,14,10,9)(8,13,12,11);(5,6)(7,12,10,8)(9,11,14,13)27 conjugacy classes — view the full character table.
| maximal subgroup of | #3715, #3717, #3718, … |
| maximal normal subgroup of | #3715, #3717, #3718, … |
| maximal quotient of | C3×Dic3⋊C4, C3×C4⋊Dic3, C2×C3×Dic6, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 26);