Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×C19×#350 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C19 | D4 |
| commutator | C4 | C2×C4×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C2×C22⋊C8 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55);(5,6,9,14,10,15,24,32,11,16,25,33,8,13,22,30)(7,17,19,34,18,28,35,31,20,29,23,36,21,12,26,27);(5,7)(6,12)(8,21)(9,23)(10,18)(11,20)(13,29)(14,31)(15,17)(16,28)(19,25)(22,35)(24,26)(27,33)(30,34)(32,36)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5591);