Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C2×C19×#1426 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #367 |
| commutator | C16 | C22×C4×C19 |
| Frattini, cosocle | C2×C16 | C23×C19 |
| prime radical | C23×C19 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C22×C8×C19 ◁ C22×C16×C19 ◁ G | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53);(4,9,11,22)(5,14)(6,18)(7,19)(10,25)(12,26,29,34)(13,32,30,24)(15,20)(16,28)(17,27)(21,33,31,23);(3,4,10,26,6,12,5,9,19,30,14,23,15,21,25,32,8,11,28,34,20,29,17,22,7,13,27,33,18,31,16,24)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6625);