Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C19×#2240 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| prime radical | C2×C19×#493 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(3,4)(5,8)(6,7)(9,10)(12,18)(13,23)(14,25)(15,26)(16,19)(20,21)(22,24);(4,7)(8,10)(11,12,15,20,16,21,14,18,17,22,25,13,19,23,26,24);(3,5,6,9)(4,8,7,10)(11,13,14,12,19,18,15,23,17,20,26,22,16,24,25,21)1444 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26682);