Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C2×C19×#2222 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| prime radical | C2×#41×C19 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(3,4,10,16)(5,17,23,45)(6,24,27,41)(7,28,30,39)(8,15,9,14)(11,32,36,20)(12,31,38,26)(13,29,40,18)(19,48,51,65)(21,44,22,43)(25,56,57,37)(33,60,61,66)(34,52,35,46)(42,53,64,50)(47,55,58,63)(49,54,59,62);(3,5,18,46,9,22,30,32,10,23,29,52,8,21,7,20)(4,11,28,44,15,35,40,45,16,36,39,43,14,34,13,17)(6,19,47,66,26,50,57,59,27,51,58,60,31,53,25,49)(12,33,55,48,24,54,56,64,38,61,63,65,41,62,37,42);(3,6)(4,12)(5,19)(7,25)(8,31)(9,26)(10,27)(11,33)(13,37)(14,41)(15,24)(16,38)(17,42)(18,47)(20,49)(21,53)(22,50)(23,51)(28,55)(29,58)(30,57)(32,59)(34,62)(35,54)(36,61)(39,63)(40,56)(43,65)(44,48)(45,64)(46,66)(52,60)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26603);