Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C3×C4×#41 (order 384) · Inn(G) = C22 · Out(G) = C23×C3×C4 |
| homology | Abelianization (H1) = C2×C8×C13 · Schur multiplier (H2) = C1 |
| direct product | C13×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | C22 |
| commutator | C2 | C2×C8×C13 |
| Frattini, cosocle | C8 | C22×C13 |
| socle | C2×C13 | C2×C8 |
| prime radical | C22×C13 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ G | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 2 generators on 29 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);(15,23)(17,25)(19,27)(21,29)260 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 60);