Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C19 · Schur multiplier (H2) = C4 |
| direct product | C19×#7197 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #95 |
| commutator | C23 | C4×C8×C19 |
| Frattini, cosocle | C2×C4×C8 | C22×C19 |
| socle | C22×C19 | #336 |
| prime radical | C23×C19 | C4×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 2 generators on 67 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,28,29,25)(23,27)(24,26)(30,34,35,32)(31,33)(36,37,39,43,40,45,50,57,41,44,51,56,52,58,63,66)(38,46,48,59,49,60,62,67,47,42,53,54,61,55,64,65);(20,22,23,30,24,29,31,35)(21,25,27,32,26,28,33,34)(36,38)(37,42,44,46)(39,48)(40,49)(41,47)(43,54,56,59)(45,55,58,60)(50,62)(51,53)(52,61)(57,65,66,67)(63,64)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 185);