Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C83 · Schur multiplier (H2) = C25 |
| direct product | C83×#33855 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C83 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C83 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C83 ◁ C2×C4×C83 ◁ #443×C83 ◁ G | |
640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 4 generators on 147 points:
(1,2,8,14)(3,15,20,48)(4,21,24,44)(5,25,30,42)(6,13,7,12)(9,33,37,18)(10,31,39,23)(11,28,43,29)(16,49,53,35)(17,54,57,34)(19,58,32,40)(22,60,55,38)(26,45,61,36)(27,41,62,64)(46,52,63,51)(47,50,59,56)(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);(1,3,8,20)(2,9,14,37)(4,16,24,53)(5,26,30,61)(6,19,7,32)(10,34,39,54)(11,40,43,58)(12,36,13,45)(15,42,48,25)(17,55,57,22)(18,28,33,29)(21,59,44,47)(23,51,31,52)(27,50,62,56)(35,60,49,38)(41,46,64,63);(1,4)(2,10)(3,16)(5,22)(6,31)(7,23)(8,24)(9,34)(11,38)(12,44)(13,21)(14,39)(15,46)(17,26)(18,50)(19,52)(20,53)(25,41)(27,29)(28,62)(30,55)(32,51)(33,56)(35,40)(36,47)(37,54)(42,64)(43,60)(45,59)(48,63)(49,58)(57,61);(1,4)(2,10)(3,17)(5,27)(6,23)(7,31)(8,24)(9,35)(11,41)(12,21)(13,44)(14,39)(15,47)(16,26)(18,52)(19,50)(20,57)(22,29)(25,38)(28,55)(30,62)(32,56)(33,51)(34,40)(36,46)(37,49)(42,60)(43,64)(45,63)(48,59)(53,61)(54,58)3071 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 26843);