Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C83 · Schur multiplier (H2) = C29 |
| direct product | C2×C4×AGL(1,ℤ/8)×C83 |
| 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 | C22×AGL(1,ℤ/8)×C83 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C83 ◁ C22×C42×C83 ◁ G | |
3242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 5 generators on 97 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(3,4,5,6);(7,8)(9,11)(10,12)(13,14);(8,11)(9,14)(12,13);(8,12)(11,13)7304 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 53144);