Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C83 · Schur multiplier (H2) = C23×C4 |
| direct product | C83×#21663 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C83 | C22×D4 |
| commutator | C2×C4 | C23×C4×C83 |
| Frattini, cosocle | C22×C4 | C24×C83 |
| socle | C22×C83 | C2×C4×D4 |
| prime radical | #532×C83 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C83 ◁ C2×C4×C8×C83 ◁ G | |
730 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)(9,10,14,21,16,22,15,20)(11,23,26,17,28,36,27,37)(12,29,13,19,32,38,31,18)(24,34,25,40,30,35,33,39)(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);(2,5)(6,8)(9,11,15,27,16,28,14,26)(10,17,21,36,22,37,20,23)(12,30,31,25,32,24,13,33)(18,35,29,39,19,34,38,40);(1,3,4,7)(2,6,5,8)(9,12,14,13,16,32,15,31)(10,18,20,38,22,19,21,29)(11,24,26,25,28,30,27,33)(17,34,23,39,37,35,36,40);(9,13,14,32,16,31,15,12)(10,19,21,38,22,18,20,29)(11,25,26,30,28,33,27,24)(17,35,36,39,37,34,23,40)5312 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 14651);