Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#21178 |
| 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 | |
866 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)(24,25,29,33,30,34,28,42)(26,35,37,31,38,45,43,46)(27,32,41,48,39,47,40,52)(36,49,51,44,50,55,53,54);(24,26,29,37,30,38,28,43)(25,31,33,45,34,46,42,35)(27,36,41,51,39,50,40,53)(32,44,48,55,47,54,52,49);(20,21,22,23)(24,27)(25,32)(26,36)(28,40)(29,41)(30,39)(31,44)(33,48)(34,47)(35,49)(37,51)(38,50)(42,52)(43,53)(45,55)(46,54);(27,39)(32,47)(36,50)(40,41)(44,54)(48,52)(49,55)(51,53)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 14166);