Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#12668 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #106 |
| commutator | C8 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | C4×#106 |
| prime radical | C23×C19 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
308 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)(20,21,25,31,26,33,39,46,27,32,24,30,38,45,23,29)(22,34,35,44,36,47,40,43,37,50,51,49,42,48,41,28);(20,22)(21,28,32,43)(23,35)(24,40)(25,41)(26,42)(27,37)(29,34,46,50)(30,47,31,48)(33,49,45,44)(36,38)(39,51);(20,23,38,24,27,39,26,25)(21,29,45,30,32,46,33,31)(22,35,36,40,37,51,42,41)(28,34,44,47,43,50,49,48)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5656);