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) = C24 |
| direct product | C19×#12547 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C42 |
| commutator | C4 | C2×C4×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C2×#97 |
| prime radical | C24×C19 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 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)(24,25,28,33,29,35,43,52,30,34,44,51,27,32,41,49)(26,36,39,53,40,31,45,47,37,48,42,55,38,46,54,50);(21,23)(24,26)(25,31,34,46)(27,38)(28,42)(29,40)(30,37)(32,36,35,48)(33,50,51,47)(39,44)(41,45)(43,54)(49,55,52,53);(26,37)(31,46)(36,48)(38,40)(39,42)(45,54)(47,50)(53,55)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5535);