Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#20807 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle | C22×C19 | C22×C4⋊C4 |
| prime radical | C24×C19 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C19×#117 ◁ G | |
954 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)(21,23)(24,25,29,33,30,34,28,43)(26,35,38,45,39,49,37,31)(27,32,42,47,40,46,41,53)(36,48,52,54,50,55,51,44);(24,26,28,37,30,39,29,38)(25,31,33,35,34,45,43,49)(27,36,41,51,40,50,42,52)(32,44,47,48,46,54,53,55);(20,21,22,23)(24,27)(25,32)(26,36)(28,41)(29,42)(30,40)(31,44)(33,47)(34,46)(35,48)(37,51)(38,52)(39,50)(43,53)(45,54)(49,55);(27,40)(32,46)(36,50)(41,42)(44,54)(47,53)(48,55)(51,52)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 13795);