Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#11620 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C22×C19 | C2×C22⋊C8 |
| prime radical | C19×#121 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 2432 |
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)(20,21,24,27,25,22,23,26)(28,29,32,38,33,39,47,56,34,36,48,52,31,37,45,54)(30,40,43,57,42,51,58,55,41,53,46,59,44,35,49,50);(20,22)(21,25)(23,27)(24,26)(28,30)(29,35)(31,44)(32,46)(33,42)(34,41)(36,51)(37,53)(38,55)(39,40)(43,48)(45,58)(47,49)(50,52)(54,57)(56,59);(20,23,25,24)(21,26,22,27)(29,36)(30,41)(37,39)(38,52)(42,44)(43,46)(49,58)(54,56)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4608);