Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C26 |
| direct product | C2×C19×#2145 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #102 |
| commutator | C8 | C23×C4×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C22×C19 | C2×#102 |
| prime radical | C22×C19×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ C23×C8×C19 ◁ G | |
1170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53);(3,4)(5,14,17,31)(6,10)(7,21)(8,13)(9,16,25,33)(11,19)(12,20)(15,30,18,27)(22,29,34,24)(23,26)(28,32);(3,5)(4,9)(6,15)(7,22)(8,17)(10,24)(11,27)(12,29)(13,25)(14,26)(16,32)(18,20)(19,33)(21,31)(23,34)(28,30);(3,6,8,20)(4,10,13,12)(5,15,17,18)(7,19,23,32)(9,24,25,29)(11,26,28,21)(14,30,31,27)(16,22,33,34)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26375);