Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13123 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #121 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1923 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ G | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 83 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,31,26,30,46,49,27,32,47,56,24,45,34,60)(22,33,37,58,38,52,67,77,39,59,68,80,36,28,48,51)(23,40,43,71,44,70,72,81,42,29,53,55,41,54,61,74)(35,62,65,82,66,50,75,76,64,57,78,79,63,69,73,83);(20,22,34,48,24,36,47,68,27,39,46,67,26,38,25,37)(21,28,49,58,45,59,31,51,32,52,60,80,30,33,56,77)(23,35,61,73,41,63,53,78,42,64,72,75,44,66,43,65)(29,50,74,79,70,62,55,76,40,69,81,82,54,57,71,83);(20,23,24,41,27,42,26,44)(21,29,30,54,32,40,45,70)(22,35,36,63,39,64,38,66)(25,43,34,61,47,53,46,72)(28,50,33,57,52,69,59,62)(31,55,49,74,56,71,60,81)(37,65,48,73,68,78,67,75)(51,76,58,79,77,83,80,82)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6111);