Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C19 · Schur multiplier (H2) = C27 |
| direct product | C19×#20351 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C4 | C22×C42×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C22×C4⋊C4 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×#41×C19 ◁ G | |
1978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,23)(24,25,28,34,30,35,29,33)(26,36,38,31,40,45,39,46)(27,32,42,49,41,47,43,48)(37,50,52,44,51,55,53,54);(21,23)(24,26,29,39,30,40,28,38)(25,31,34,45,35,46,33,36)(27,37,43,53,41,51,42,52)(32,44,49,55,47,54,48,50);(24,27)(25,32)(26,37)(28,42)(29,43)(30,41)(31,44)(33,48)(34,49)(35,47)(36,50)(38,52)(39,53)(40,51)(45,55)(46,54);(27,41)(32,47)(37,51)(42,43)(44,54)(48,49)(50,55)(52,53)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13339);