Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C83 · Schur multiplier (H2) = C27×C4 |
| direct product | C83×#60129 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C83 | C25 |
| commutator | C22 | C24×C4×C83 |
| Frattini, cosocle | C2×C4 | C25×C83 |
| socle | C22×C83 | C24×C4 |
| prime radical | C22×#41×C83 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C83 ◁ G | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 5 generators on 123 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,17,14,18,24,30)(11,15,21,27,19,25,32,36)(12,16,23,29,22,28,34,38)(20,26,33,37,31,35,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,26)(17,27)(18,25)(22,31)(23,33)(24,32)(28,35)(29,37)(30,36)(34,39)(38,40);(1,3,4,7)(2,6,5,8)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40);(9,12)(10,16)(11,20)(13,23)(14,22)(15,26)(17,29)(18,28)(19,31)(21,33)(24,34)(25,35)(27,37)(30,38)(32,39)(36,40);(12,22)(16,28)(20,31)(23,34)(26,35)(29,38)(33,39)(37,40)7304 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 53117);