Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C131 · Schur multiplier (H2) = C23 |
| direct product | C131×#13418 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | #2041 |
| commutator, Frattini, cosocle | C22×C8 | C23×C131 |
| prime radical | C2×#135×C131 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C22×C4×C131 ◁ #405×C131 ◁ G | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 195 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 3 generators on 195 points:
(1,2)(3,17)(4,24)(5,15)(6,31)(7,13)(8,12)(9,16)(10,19)(11,27)(14,33)(18,50)(20,37)(21,38)(22,40)(23,39)(25,36)(26,43)(28,48)(29,61)(30,45)(32,42)(34,44)(35,60)(41,53)(46,54)(47,56)(49,52)(51,62)(55,63)(57,59)(58,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195);(1,3,9,23)(2,10,16,40)(4,25,29,55)(5,22,8,19)(6,21,33,54)(7,20,30,56)(11,41,44,50)(12,39,15,17)(13,46,45,38)(14,47,31,37)(18,32,53,28)(24,59,61,49)(26,52,35,57)(27,51,34,58)(36,60,63,43)(42,64,48,62);(1,4,5,26,9,29,8,35)(2,11,12,42,16,44,15,48)(3,18,19,51,23,53,22,58)(6,32,30,27,33,28,7,34)(10,36,17,49,40,63,39,59)(13,43,14,24,45,60,31,61)(20,55,54,52,56,25,21,57)(37,41,38,62,47,50,46,64)4454 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 6406);