Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12915 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #120 |
| commutator | C23 | C22×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1850 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ G | |
374 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)(20,21,24,29,25,30,39,49,26,31,40,50,41,51,62,72)(22,27,34,44,35,45,58,68,33,43,56,66,57,67,77,81)(23,28,37,47,38,48,61,71,36,46,59,69,60,70,78,82)(32,52,54,73,55,74,76,83,53,42,63,64,75,65,79,80);(20,22,24,34,25,35,39,58,26,33,40,56,41,57,62,77)(21,27,29,44,30,45,49,68,31,43,50,66,51,67,72,81)(23,32,37,54,38,55,61,76,36,53,59,63,60,75,78,79)(28,42,47,64,48,65,71,80,46,52,69,73,70,74,82,83);(20,23)(21,28)(22,32)(24,37)(25,38)(26,36)(27,42)(29,47)(30,48)(31,46)(33,53)(34,54)(35,55)(39,61)(40,59)(41,60)(43,52)(44,64)(45,65)(49,71)(50,69)(51,70)(56,63)(57,75)(58,76)(62,78)(66,73)(67,74)(68,80)(72,82)(77,79)(81,83)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5903);