Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C83 · Schur multiplier (H2) = C210 |
| direct product | C2×C83×#2896 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C83 | C2×D4 |
| commutator | C4 | C24×C4×C83 |
| Frattini, cosocle | C2×C4 | C25×C83 |
| socle | C23×C83 | C22×D4 |
| prime radical | C23×#41×C83 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C83 ◁ C24×C4×C83 ◁ G | |
5198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 5 generators on 105 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(3,4,5,6)(7,8)(9,16)(10,15)(11,19)(12,14)(13,17)(18,22)(20,21);(7,9)(8,13)(10,17)(11,20)(12,18)(14,21)(15,22)(16,19);(8,14)(13,21)(15,19)(16,22);(7,10,12,11)(8,15,14,19)(9,17,18,20)(13,22,21,16)7304 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 53071);