Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#125 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C13 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C13 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29);(1,3,4,7)(2,5,6,8)(10,13)(14,16);(2,6)(5,8)(9,11,12,15)(10,14,13,16)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 186);