Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 18432 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C22×C4 |
| abelian | C2×C42×C13 |
| multiplicative | (ℤ/795)ˣ, (ℤ/848)ˣ, (ℤ/1060)ˣ, (ℤ/1590)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C22 | C23×C13 |
| socle, prime radical | C23×C13 | C22 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4)(11,12,13,14,15,16,17,18,19,20,21,22,23);(5,6,7,8);(9,10)416 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 175);