Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C23 |
| direct product | C2×C19×#1332 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C19 | D8 |
| commutator | C8 | C22×C8×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle, prime radical | C23×C19 | #102 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 4864 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(3,4,5,7,6,8,9,10)(12,16)(13,19)(14,21)(17,26)(18,25)(20,23)(22,24);(11,12,15,18)(13,20,22,26)(14,23,21,17)(16,24,25,19)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4578);