Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C131 · Schur multiplier (H2) = C2×C4 |
| direct product | C131×#13271 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C131 | #476 |
| commutator | C2×C8 | C22×C4×C131 |
| Frattini, cosocle | C22×C8 | C23×C131 |
| socle | C2×C131 | #1987 |
| prime radical | C23×C131 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C131 ◁ C22×C4×C131 ◁ C2×C4×C8×C131 ◁ G | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 4192 |
A minimal faithful permutation representation — 3 generators on 163 points:
(1,2,5,11,7,12,21,31,8,13,20,10,6,23,4,19)(3,14,15,26,17,30,32,9,18,27,16,28,24,25,22,29)(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)(2,9,13,29)(4,15)(5,22)(6,17)(7,24)(8,18)(10,25,11,30)(12,26,23,28)(14,19,27,31)(16,21)(20,32);(1,4,6,20,8,21,7,5)(2,10,12,19,13,11,23,31)(3,15,17,32,18,16,24,22)(9,25,26,27,29,30,28,14)6026 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 6259);