Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#17409 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C2×C19×#121 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C4×Q8×C19 ◁ G | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,27,24,26,30,33)(22,25,29,32,28,31,34,35)(36,37,41,44,42,45,40,49)(38,46,47,50,39,43,48,51);(20,22)(21,25)(23,29)(24,28)(26,31)(27,32)(30,34)(33,35)(36,38)(37,43)(39,42)(40,47)(41,48)(44,50)(45,46)(49,51);(22,28)(25,31)(29,34)(32,35)(36,39,42,38)(37,43,45,46)(40,47,41,48)(44,51,49,50);(21,26)(25,31)(27,33)(32,35)(36,40,42,41)(37,44,45,49)(38,47,39,48)(43,50,46,51)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10397);