Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#17837 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C22×C4⋊C4 |
| prime radical | C2×#41×C19 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C4×#41×C19 ◁ G | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,26,23)(22,29,33,35)(24,28,25,34)(27,31,30,32)(36,37,40,44,41,45,39,43)(38,46,48,51,49,50,47,42);(20,22,24,31,26,33,25,32)(21,27,34,35,23,30,28,29)(36,38,40,48,41,49,39,47)(37,42,43,50,45,51,44,46);(20,23,26,21)(22,30,33,27)(24,28,25,34)(29,32,35,31)(36,39,41,40)(37,43,45,44)(38,47,49,48)(42,50,51,46);(20,24,26,25)(21,28,23,34)(22,31,33,32)(27,29,30,35)(36,39,41,40)(37,43,45,44)(38,47,49,48)(42,50,51,46)1330 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10825);