Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C83 · Schur multiplier (H2) = C26 |
| direct product | C83×#22460 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C83 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C83 |
| Frattini, cosocle | C22×C4 | C24×C83 |
| prime radical | C2×#578×C83 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C83 ◁ C22×C4×C83 ◁ G | |
1286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 4 generators on 147 points:
(1,2,7,12,8,13,6,26)(3,14,17,9,18,33,27,32)(4,19,21,35,22,10,29,36)(5,11,24,40,25,39,28,38)(15,43,46,30,47,41,45,50)(16,42,48,31,49,37,23,56)(20,51,54,60,55,34,53,59)(44,58,52,57,62,61,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(1,3)(2,9)(4,15)(5,23)(6,27)(7,17)(8,18)(10,30)(11,37)(12,33)(13,32)(14,26)(16,24)(19,50)(20,52)(21,46)(22,47)(25,48)(28,49)(29,45)(31,38)(34,58)(35,43)(36,41)(39,42)(40,56)(44,53)(51,61)(54,62)(55,63)(57,59)(60,64);(1,4,8,22)(2,10,13,19)(3,15,18,47)(5,20,25,55)(6,21,7,29)(9,30,32,50)(11,34,39,51)(12,35,26,36)(14,41,33,43)(16,44,49,62)(17,45,27,46)(23,52,48,63)(24,53,28,54)(31,57,56,64)(37,58,42,61)(38,59,40,60);(1,5)(2,11)(3,16)(4,20)(6,28)(7,24)(8,25)(9,31)(10,34)(12,40)(13,39)(14,42)(15,44)(17,48)(18,49)(19,51)(21,54)(22,55)(23,27)(26,38)(29,53)(30,57)(32,56)(33,37)(35,60)(36,59)(41,61)(43,58)(45,63)(46,52)(47,62)(50,64)4316 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 15448);