Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#12602 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C22⋊C4 |
| commutator | C22 | C2×C4×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C2×C22⋊C8 |
| prime radical | C24×C19 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22)(24,25)(26,34)(27,38)(28,33)(29,31)(30,35)(32,40)(36,42)(37,47)(39,51)(41,44)(43,49)(45,52)(46,48)(50,53)(54,55);(24,26,27,36,28,37,39,50,29,35,40,48,41,49,52,55)(25,30,32,42,33,43,45,53,31,34,38,46,44,47,51,54);(20,22,23,21)(25,31)(30,34)(32,38)(33,44)(42,46)(43,47)(45,51)(53,54)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5590);