Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C83 · Schur multiplier (H2) = C24×C4 |
| direct product | C83×#17366 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C83 | C2×C22⋊C4 |
| commutator | C23 | C23×C4×C83 |
| Frattini, cosocle | C22×C4 | C24×C83 |
| socle | C22×C83 | C22×C22⋊C4 |
| prime radical | C22×#41×C83 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C83 ◁ C22×C42×C83 ◁ G | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 4 generators on 107 points:
(1,2,4,8,5,6,3,7)(9,10)(11,19,22,15)(12,16)(13,17)(14,18)(20,24,21,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,17,18,16)(11,21,22,20)(15,24,19,23)5312 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 10354);