Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C83 · Schur multiplier (H2) = C24 |
| direct product | C83×#12979 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C83 | #120 |
| commutator | C2×C4 | C22×C8×C83 |
| Frattini, cosocle | C22×C8 | C23×C83 |
| socle | C2×C83 | #1869 |
| prime radical | #575×C83 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C83 ◁ C22×C8×C83 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 3 generators on 147 points:
(1,2,6,12,7,13,27,40,8,14,28,41,5,11,25,38)(3,9,17,31,16,30,47,58,18,32,49,60,23,36,53,62)(4,19,22,51,21,34,52,39,20,37,26,54,24,10,33,35)(15,42,45,63,46,29,55,57,43,50,48,64,44,56,61,59)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(1,3,6,17,7,16,27,47,8,18,28,49,5,23,25,53)(2,9,12,31,13,30,40,58,14,32,41,60,11,36,38,62)(4,15,26,48,21,46,33,61,20,43,22,45,24,44,52,55)(10,29,39,59,19,50,54,63,34,56,35,57,37,42,51,64);(1,4)(2,10)(3,15)(5,24)(6,26)(7,21)(8,20)(9,29)(11,37)(12,39)(13,19)(14,34)(16,46)(17,48)(18,43)(22,28)(23,44)(25,52)(27,33)(30,50)(31,59)(32,56)(35,41)(36,42)(38,51)(40,54)(45,49)(47,61)(53,55)(57,60)(58,63)(62,64)5312 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 5967);