Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C8 |
| direct product | C19×#8270 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C19 | C2×D4 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | C4×C4⋊C4 |
| prime radical | C23×C19 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1216 |
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)(21,25,27,30)(26,32,33,34)(36,37)(38,41)(39,43)(40,42)(44,48)(45,47)(46,49)(50,51);(20,21)(22,26)(23,30)(24,27)(25,31)(28,34)(29,33)(32,35)(36,38,39,44,40,45,46,50)(37,41,43,48,42,47,49,51);(20,22,23,28,24,29,31,35)(21,26,30,34,27,33,25,32)(37,42)(41,47)(43,49)(48,51)1444 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1258);