Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#17924 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C22×D4×C19 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C19×#431 ◁ G | |
638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1216 |
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)(20,21,24,27)(22,26,29,34)(23,25,31,28)(30,32,35,33)(36,37,39,43,40,42,46,49)(38,41,44,48,45,47,50,51);(21,25)(23,31)(26,32)(27,28)(30,35)(33,34);(20,22)(21,26)(23,30)(24,29)(25,32)(27,34)(28,33)(31,35)(36,38,39,44,40,45,46,50)(37,41,43,48,42,47,49,51);(21,27)(22,29)(25,28)(30,35)(37,42)(41,47)(43,49)(48,51)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10912);