Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C83 · Schur multiplier (H2) = C24 |
| direct product | C2×C83×#1378 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C83 | C22⋊C4 |
| commutator | C22 | C2×C4×C8×C83 |
| Frattini, cosocle | C22×C8 | C23×C83 |
| socle | C22×C83 | #336 |
| prime radical | C24×C83 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C83 ◁ C23×C8×C83 ◁ G | |
586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(3,4,6,10,7,12,17,24,8,11,18,23,19,25,30,33)(5,13,15,26,16,27,29,34,14,9,20,21,28,22,31,32);(3,5)(4,9,11,13)(6,15)(7,16)(8,14)(10,21,23,26)(12,22,25,27)(17,29)(18,20)(19,28)(24,32,33,34)(30,31)7304 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 5540);