Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C83 · Schur multiplier (H2) = C24 |
| direct product | C83×#8349 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C83 | #379 |
| commutator | C22×C4 | C22×C4×C83 |
| Frattini, cosocle | C2×C42 | C23×C83 |
| prime radical | C83×#3493 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 3 generators on 123 points:
(2,5,6,7)(9,10,13,18,14,17,26,31)(11,19,22,32,21,15,25,28)(12,16,24,29,23,30,36,40)(20,27,35,38,34,33,37,39)(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);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,20)(13,25)(14,21)(16,27)(17,19)(18,32)(22,26)(23,34)(24,37)(28,31)(29,39)(30,33)(35,36)(38,40);(9,12,14,23)(10,16,17,30)(11,20,21,34)(13,24,26,36)(15,27,19,33)(18,29,31,40)(22,35,25,37)(28,38,32,39)3818 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 1337);