Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13236 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C19 | C2×D8 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 4864 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(24,25,26,27)(28,29)(30,41)(31,47)(32,50)(33,51)(34,40)(35,43)(36,44)(37,49)(38,42)(39,48)(45,59)(46,52)(53,56)(54,58)(55,57);(20,21,22,23)(25,27)(28,30,42,56,33,45,31,43,34,46,49,58,48,57,32,44)(29,35,50,59,39,53,37,41,40,36,47,55,51,54,38,52);(28,30,42,56,33,45,31,43,34,46,49,58,48,57,32,44)(29,36,50,55,39,54,37,52,40,35,47,59,51,53,38,41)1444 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6224);