Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C83 · Schur multiplier (H2) = C2 |
| direct product | C83×#7460 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C83 | D16 |
| commutator | C16 | C2×C8×C83 |
| Frattini, cosocle | C4×C16 | C22×C83 |
| socle, prime radical | C22×C83 | #366 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C83 ◁ C42×C83 ◁ C4×C8×C83 ◁ C4×C16×C83 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 21248 |
A minimal faithful permutation representation — 2 generators on 123 points:
(1,2,3,5,4,6,7,8)(10,15)(11,16)(12,20)(13,23)(17,32)(18,35)(19,28)(21,37)(22,31)(24,36)(25,26)(27,33)(29,30)(34,40)(38,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(9,10,16,26,12,17,29,35,23,33,37,40,36,38,22,28,14,19,31,39,24,34,21,27,13,18,30,32,20,25,11,15)6308 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 448);