Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#33713 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C22×C19 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 2432 |
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)(26,35)(27,32)(28,42)(29,44)(30,34)(31,38)(33,36)(37,48)(39,54)(40,50)(41,47)(43,55)(45,52)(46,53)(49,51);(24,26,36,51,29,39,43,53,30,40,41,52,27,37,28,38)(25,31,42,48,32,45,47,50,34,46,55,54,44,49,33,35);(20,21,22,23);(24,27,30,29)(25,32,34,44)(26,37,40,39)(28,41,43,36)(31,45,46,49)(33,42,47,55)(35,48,50,54)(38,52,53,51)1444 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26701);