Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C19×#1659 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C2×C19×H(3,𝔽4) | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C2×C42×C19 ◁ G | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(4,7)(5,9)(8,10)(11,12,16,20)(13,21,23,26)(14,19,15,18)(17,24,25,22);(5,9)(7,10)(11,13)(12,17)(14,24)(15,22)(16,23)(18,26)(19,21)(20,25);(3,4)(5,10)(6,8)(7,9)(11,13)(12,17)(14,22)(15,24)(16,23)(18,21)(19,26)(20,25)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 11967);