Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#12665 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C2×D4 |
| commutator | C4 | C2×C4×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C8×D4 |
| prime radical | C23×C19 | #131 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C22×C16×C19 ◁ G | |
326 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)(24,25,29,33,28,32,41,27,30,34,46,42,44,51,55,43)(26,35,38,36,39,31,47,48,40,49,45,54,37,50,53,52);(20,21,22,23)(24,26)(25,31)(27,36)(28,39)(29,45)(30,40)(32,49)(33,52)(34,50)(35,51)(37,44)(38,46)(41,53)(42,48)(43,54)(47,55);(24,27,41,51,44,33,29,34,30,43,55,32,28,42,46,25)(26,36,53,35,37,52,45,50,40,54,47,49,39,48,38,31)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5653);