Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13161 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C2×D4 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C8×D4 |
| prime radical | C23×C19 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C22×C16×C19 ◁ G | |
298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,23)(24,25,29,35,30,34,44,50,31,36,43,33,28,45,27,42)(26,32,39,48,40,47,53,55,41,49,52,46,38,54,37,51);(21,23)(24,26,29,39,30,40,44,53,31,41,43,52,28,38,27,37)(25,32,35,48,34,47,50,55,36,49,33,46,45,54,42,51);(24,27,28,43,31,44,30,29)(25,33,34,42,36,35,45,50)(26,37,38,52,41,53,40,39)(32,46,47,51,49,48,54,55)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6149);