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×#21429 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C83 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C83 |
| Frattini, cosocle | C22×C4 | C24×C83 |
| prime radical | C24×C83 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C83 ◁ C2×#117×C83 ◁ G | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 4 generators on 123 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,18,14,17,19,31)(11,15,22,27,21,28,25,32)(12,16,24,30,23,29,33,39)(20,26,35,38,34,37,36,40)(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);(9,11,19,22,14,21,13,25)(10,15,18,32,17,28,31,27)(12,20,33,35,23,34,24,36)(16,26,30,40,29,37,39,38);(2,5)(6,8)(9,12,14,23)(10,16,17,29)(11,20,21,34)(13,24,19,33)(15,26,28,37)(18,30,31,39)(22,35,25,36)(27,38,32,40);(1,3,4,7)(2,6,5,8)(12,23)(16,29)(20,34)(24,33)(26,37)(30,39)(35,36)(38,40)4814 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 14417);