Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12617 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #117 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1752 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C2×C19×M32 ◁ G | |
382 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)(20,21,25,31,26,32,47,61,27,33,48,62,24,30,45,59)(22,34,37,63,36,52,66,60,38,58,46,74,44,28,49,51)(23,29,41,55,43,57,73,82,42,56,72,81,40,54,70,79)(35,53,64,77,39,67,68,83,65,78,75,80,69,50,71,76);(20,22)(21,28)(23,39,42,69)(24,44)(25,46)(26,36)(27,38)(29,53,56,78)(30,58)(31,60)(32,34)(33,52)(35,43,65,40)(37,48)(41,71,72,68)(45,66)(47,49)(50,57,67,54)(51,62)(55,80,81,77)(59,63)(61,74)(64,70,75,73)(76,79,83,82);(20,23,25,41,26,43,47,73,27,42,48,72,24,40,45,70)(21,29,31,55,32,57,61,82,33,56,62,81,30,54,59,79)(22,35,37,64,36,39,66,68,38,65,46,75,44,69,49,71)(28,50,51,76,34,53,63,77,52,67,60,83,58,78,74,80)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5605);