Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#7174 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #106 |
| commutator | C8 | C4×C8×C19 |
| Frattini, cosocle | C2×C4×C8 | C22×C19 |
| socle | C22×C19 | #340 |
| prime radical | C23×C19 | #102 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,26,24,27,22,25)(28,29,32,35,34,38,42,37)(30,33,39,43,36,31,40,41);(21,25,27,26)(28,30,33,35,32,40,43,29,34,36,31,37,42,39,41,38)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 162);