Finite Group DatabasebetaFinite Group Databasebeta| order | 222 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C4×C9 (order 72) · Inn(G) = C1 · Out(G) = C2×C4×C9 |
| holomorph | G ⋊ Aut(G) = C2×D3×AGL(1,𝔽37) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C222 |
| abelian | C2×C3×C37 |
| multiplicative | (ℤ/223)ˣ, (ℤ/446)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 222 |
A minimal faithful permutation representation — 1 generator on 42 points:
(1,2)(3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)222 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(222, 6);