Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C131 · Schur multiplier (H2) = C25 |
| direct product | C131×#8468 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | #379 |
| commutator | C22×C4 | C22×C4×C131 |
| Frattini, cosocle | C23×C4 | C23×C131 |
| prime radical | #525×C131 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C23×C4×C131 ◁ G | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 3 generators on 163 points:
(1,2)(3,9,11,6)(4,12,13,7)(5,8)(10,14)(15,16)(18,22,24,27)(23,31,32,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,18)(19,23)(20,27)(21,24)(22,29)(25,30)(26,32)(28,31);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,19,21,26)(18,23,24,32)(20,28,29,25)(22,30,27,31)5240 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 1456);