Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12618 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1752 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C2×C19×M32 ◁ G | |
382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 608 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,28,24,29,34,42,25,26,33,37,35,38,47,49)(22,27,31,40,30,41,43,51,32,36,46,48,44,39,45,50);(21,26)(22,30,32,44)(23,33)(27,39,36,41)(29,38)(31,45,46,43)(34,47)(40,51,48,50);(20,22,23,31,24,30,34,43,25,32,33,46,35,44,47,45)(21,27,28,40,29,41,42,51,26,36,37,48,38,39,49,50)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5606);