Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#33864 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C4×D4×C19 ◁ C8×D4×C19 ◁ G | |
580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,25)(22,26,27,24)(28,29,33,38)(30,34,41,49)(31,35,44,51)(32,45,43,37)(36,52,50,42)(39,46,55,58)(40,56,54,48)(47,59,57,53);(20,22)(21,24)(23,27)(25,26)(28,30,33,41)(29,34,38,49)(31,39,44,55)(32,40,43,54)(35,46,51,58)(36,47,50,57)(37,48,45,56)(42,53,52,59);(29,35)(31,42)(32,43)(34,46)(36,37)(38,51)(39,53)(40,54)(44,52)(45,50)(47,48)(49,58)(55,59)(56,57);(22,27)(24,26)1045 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26852);