Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2 |
| direct product | C19×#9610 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #121 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | #389 |
| prime radical | C23×C19 | C4×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,22,24,26,23,27)(29,34)(30,37)(31,32)(35,42)(36,40)(38,39);(20,22,25,26,24,27,23,21)(28,29)(30,35)(31,40)(32,34)(33,36)(37,41)(38,43)(39,42);(20,23,24,25)(21,22,26,27)(28,30,31,38,33,39,32,37)(29,35,40,43,36,42,34,41)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 2598);