Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C19 · Schur multiplier (H2) = C4×C8 |
| direct product | C19×#8137 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C42 |
| commutator | C4 | C2×C4×C8×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | #376 |
| prime radical | C23×C19 | #117 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ G | |
354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,24,23,25,26,27)(28,29)(30,38,41,51)(31,35)(32,36)(33,37)(34,40,46,54)(39,50,53,58)(42,47)(43,48)(44,49)(45,52,56,59)(55,57);(28,30)(29,34,37,46)(31,39)(32,40)(33,41)(35,45,48,56)(36,38,49,51)(42,52)(43,53)(44,54)(47,50,57,58)(55,59);(28,31,32,42,33,43,44,55)(29,35,36,47,37,48,49,57)(30,39,40,52,41,53,54,59)(34,45,38,50,46,56,51,58)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1125);