Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C19 · Schur multiplier (H2) = C23 |
| direct product | C2×C19×#1370 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C19 | D4 |
| commutator | C4 | C22×C16×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle, prime radical | C23×C19 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 4864 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(4,7)(5,8)(9,10)(11,12,13,16,14,17,19,22,15,18,20,23,21,24,25,26);(3,4,8,10,6,9,5,7)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4991);