Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C16×C19 · Schur multiplier (H2) = C4 |
| direct product | C19×#7329 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C16×C19 | C22 |
| commutator | C2 | C8×C16×C19 |
| Frattini, cosocle | C4×C16 | C22×C19 |
| socle, prime radical | C22×C19 | C4×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C16×C19 ◁ G | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 4864 |
A minimal faithful permutation representation — 2 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(28,29,30,35,31,36,38,47,32,37,39,48,41,49,50,57,33,34,40,44,42,45,51,54,43,46,52,55,53,56,58,59);(20,21,22,24,23,25,26,27)(29,34)(35,44)(36,45)(37,46)(47,54)(48,55)(49,56)(57,59)3040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 317);