Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C131 · Schur multiplier (H2) = C24 |
| direct product | C131×#8501 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | #379 |
| commutator | C42 | C22×C4×C131 |
| Frattini, cosocle | C2×C42 | C23×C131 |
| prime radical | #121×C131 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C2×C42×C131 ◁ G | |
616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 195 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 3 generators on 195 points:
(1,2,5,12,8,3,14,18)(4,20,23,53,25,45,55,52)(6,11,27,39,31,17,42,49)(7,13,28,10,32,19,43,16)(9,33,35,62,15,44,34,29)(21,54,24,38,26,59,58,48)(22,36,56,41,57,46,61,51)(30,40,60,37,63,50,64,47)(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)(4,15)(5,10)(6,11)(7,19)(9,25)(12,28)(13,32)(14,16)(17,31)(18,43)(20,22)(21,34)(23,41)(24,46)(26,35)(27,47)(29,38)(30,40)(33,52)(36,58)(37,42)(39,64)(44,53)(45,57)(48,62)(49,60)(50,63)(51,55)(54,56)(59,61);(1,4,7,24)(2,9,13,36)(3,15,19,46)(5,26,28,23)(6,29,30,22)(8,25,32,58)(10,35,12,41)(11,38,40,20)(14,21,43,55)(16,34,18,51)(17,48,50,45)(27,56,60,33)(31,62,63,57)(37,59,39,53)(42,61,64,44)(47,54,49,52)4454 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 1489);