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) = C22 |
| direct product | C2×C19×#1422 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C19 | D4 |
| commutator | C4 | C22×C16×C19 |
| Frattini, cosocle | C2×C16 | C23×C19 |
| socle | C22×C19 | C4⋊C16 |
| prime radical | C23×C19 | M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C19 ◁ C22×C16×C19 ◁ G | |
214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53);(3,4,7,13,8,14,22,29,9,12,23,27,19,28,32,34,10,15,24,30,17,11,25,26,6,20,21,33,5,16,18,31);(3,5,6,17,10,19,9,8)(4,11,12,16,15,14,20,28)(7,18,21,25,24,32,23,22)(13,26,27,31,30,29,33,34)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6617);