Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C131 · Schur multiplier (H2) = C22×C4 |
| direct product | C131×#12648 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | #442 |
| commutator | C8 | C2×C42×C131 |
| Frattini, cosocle | C22×C8 | C23×C131 |
| prime radical | C23×C131 | AGL(1,ℤ/8) |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C22×C4×C131 ◁ C2×#97×C131 ◁ G | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 3 generators on 167 points:
(1,2)(3,4)(5,6,9,14)(7,17)(8,23,10,26)(11,28,24,16)(12,20)(13,27,15,25)(18,33)(19,32)(21,36)(22,30)(29,34)(31,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(2,4)(5,7,10,20,11,21,27,34,9,22,8,19,24,18,25,35)(6,12,15,30,16,31,26,36,14,32,13,17,28,29,23,33);(7,18,22,21)(8,10)(12,29,32,31)(13,15)(17,33,30,36)(19,35,20,34)(23,26)(25,27)6026 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 5636);