Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C83 · Schur multiplier (H2) = C25 |
| direct product | C83×#8488 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C83 | #379 |
| commutator | C22×C4 | C22×C4×C83 |
| Frattini, cosocle | C23×C4 | C23×C83 |
| prime radical | C83×#3535 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C83 ◁ C23×C4×C83 ◁ G | |
768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 3 generators on 123 points:
(1,2)(3,4,8,6)(5,7)(9,10,14,19,15,20,13,18)(11,21,25,16,24,31,23,30)(12,17,26,32,27,33,28,34)(22,36,37,40,38,35,39,29)(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);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,22)(13,25)(14,23)(15,24)(17,29)(18,31)(19,21)(20,30)(26,39)(27,38)(28,37)(32,35)(33,40)(34,36);(1,4,5,6)(2,3,7,8)(9,12)(10,17)(11,22)(13,28)(14,26)(15,27)(16,29)(18,34)(19,32)(20,33)(21,35)(23,39)(24,38)(25,37)(30,40)(31,36)3071 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 1476);