Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C19×C32 · Schur multiplier (H2) = C23 |
| direct product | D4×C19×C32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C19×C32 | C22 |
| commutator | C2 | C22×C19×C32 |
| Frattini, cosocle | C2×C16 | C23×C19 |
| socle | C22×C19 | C22×C16 |
| prime radical | C2×D4×C19 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19×C32 ◁ G | |
214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55);(2,4);(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)3040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6632);