Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#19124 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #121 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C2×#121 |
| prime radical | C19×#3035 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ G | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,24)(22,26)(25,27)(28,29)(30,37)(31,35)(32,40)(33,36)(34,38)(39,42)(41,43);(20,21,22,24,23,25,26,27);(28,30)(29,34)(31,38)(32,41)(33,39)(35,42)(36,43)(37,40);(28,31,33,32)(29,35,36,40)(30,38,39,41)(34,42,43,37)1102 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12112);