Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#9056 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #120 |
| commutator | C42 | C22×C4×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | #386 |
| prime radical | C23×C19 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ G | |
432 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)(22,28,29,31)(36,37)(38,46,49,50)(39,43,42,48)(40,44)(41,45)(47,51);(20,21,23,30,24,27,32,25)(22,26,31,34,29,35,28,33)(36,38,47,48,41,49,40,43)(37,42,51,46,45,39,44,50);(20,22)(21,26)(23,31)(24,29)(25,33)(27,35)(28,32)(30,34)(36,39,41,42)(37,43,45,48)(38,44,49,51)(40,46,47,50)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 2044);