Finite Group DatabasebetaFinite Group Databasebeta| order | 172 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C3×C7 (order 84) · Inn(G) = C1 · Out(G) = C22×C3×C7 |
| holomorph | G ⋊ Aut(G) = D4×AGL(1,𝔽43) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C172 |
| abelian | C4×C43 |
| multiplicative | (ℤ/173)ˣ, (ℤ/346)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C2×C43 |
| socle, prime radical | C2×C43 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
6 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 172 |
A minimal faithful permutation representation — 1 generator on 47 points:
(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47)172 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(172, 2);