Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#7470 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C19 | D4 |
| commutator | C4 | C4×C16×C19 |
| Frattini, cosocle | C22×C16 | C22×C19 |
| socle | C22×C19 | C22⋊C16 |
| prime radical | C23×C19 | M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C19 ◁ C22×C16×C19 ◁ G | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 2 generators on 87 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25,28,34,29,35,47,60,30,36,48,61,50,63,75,83,31,37,49,62,51,64,76,84,27,33,44,57,45,58,72,81)(26,38,40,65,41,66,69,85,39,55,67,59,68,79,86,82,42,56,46,73,70,80,74,87,43,32,52,53,71,54,77,78);(20,21,22,23)(24,26)(25,32)(27,43)(28,46)(29,41)(30,39)(31,42)(33,56)(34,59)(35,54)(36,38)(37,55)(40,49)(44,67)(45,71)(47,74)(48,52)(50,68)(51,70)(53,62)(57,65)(58,80)(60,82)(61,73)(63,66)(64,79)(69,76)(72,86)(75,77)(78,84)(81,85)(83,87)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 458);