Finite Group DatabasebetaFinite Group Databasebeta| order | 68 |
| isoclinism | isoclinic class of D17 |
| automorphism | Aut(G) = C2×AGL(1,𝔽17) (order 544) · Inn(G) = D17 · Out(G) = C2×C8 |
| holomorph | G ⋊ Aut(G) = #72218906 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D34 |
| direct product | C2×D17 |
| unambiguous semidirect product | C17⋊C22 |
| unambiguous semidirect product | (C2×C17)⋊C2 |
| Fitting, socle | C2×C17 ⋊ C2 |
| derived, Sylow | C17 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D17 |
| commutator, cosocle, hypocenter | C17 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C17 | C2 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C2 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 34 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19);(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)20 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D17, D68, C22⋊D17, … |
| maximal normal subgroup of | C4×D17, D68, C22⋊D17, … |
| maximal quotient of | Dic34, C4×D17, D68, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(68, 4);