Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C4×D4×C13 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C22 |
| commutator | C2 | C22×C4×C13 |
| Frattini, cosocle | C22 | C23×C13 |
| socle | C22×C13 | C23 |
| prime radical | C2×D4×C13 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ G | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4)(9,10,11,12,13,14,15,16,17,18,19,20,21);(5,6,7,8);(6,8)260 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 179);