Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C131 · Schur multiplier (H2) = C23 |
| direct product | C131×#13403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C23×C131 |
| prime radical | C2×#41×C131 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C22×C4×C131 ◁ C2×#131×C131 ◁ G | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 195 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 3 generators on 195 points:
(1,2,7,12)(3,13,18,33)(4,19,24,54)(5,25,27,10)(6,28,30,52)(8,17,34,15)(9,35,39,22)(11,40,42,26)(14,44,47,64)(16,32,51,62)(20,55,45,63)(21,53,23,41)(29,38,56,37)(31,59,61,49)(36,58,60,46)(43,48,57,50)(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)(2,8)(4,20)(5,17)(6,16)(7,18)(9,36)(10,33)(11,32)(12,34)(13,25)(14,23)(15,27)(19,44)(21,47)(22,49)(24,45)(26,50)(28,57)(29,58)(30,51)(31,38)(35,59)(37,61)(39,60)(40,48)(41,63)(42,62)(43,52)(46,56)(53,55)(54,64);(1,4,7,24)(2,9,12,39)(3,14)(5,23,27,21)(6,29,30,56)(8,31)(10,38,25,37)(11,41,42,53)(13,36)(15,20)(16,49)(17,45)(18,47)(19,52,54,28)(22,40,35,26)(32,44)(33,60)(34,61)(43,63)(46,50)(48,58)(51,59)(55,57)(62,64)4454 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 6391);