Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#12664 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C19 | D4 |
| commutator | C4 | C2×C4×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C8×D4 |
| prime radical | C23×C19 | #131 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C19 ◁ C22×C16×C19 ◁ G | |
334 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)(24,25,29,34,30,35,47,44,31,36,48,43,28,33,42,27)(26,37,40,38,39,51,54,53,41,52,46,55,45,32,49,50);(20,21,22,23)(24,26)(25,32)(27,38)(28,45)(29,46)(30,39)(31,41)(33,52)(34,53)(35,37)(36,51)(40,48)(42,54)(43,50)(44,55)(47,49);(24,27,42,33,28,43,48,36,31,44,47,35,30,34,29,25)(26,38,54,52,45,50,40,51,41,55,49,37,39,53,46,32)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5652);