Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C131 · Schur multiplier (H2) = C2×C4 |
| direct product | C131×#13267 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C131 | C2×D4 |
| commutator | C2×C4 | C22×C8×C131 |
| Frattini, cosocle | C22×C8 | C23×C131 |
| socle | C22×C131 | C2×C22⋊C8 |
| prime radical | C23×C131 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C131 ◁ C2×C4×C8×C131 ◁ G | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 171 |
| transitive degree | μt(G) = 16768 |
A minimal faithful permutation representation — 3 generators on 171 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,18,14,17,27,30,15,19,28,33,12,26,21,36)(11,20,24,35,25,31,38,39,22,34,37,40,23,16,29,32)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171);(2,5)(6,7)(9,11,21,29,12,23,28,37,15,22,27,38,14,25,13,24)(10,16,30,35,26,34,18,32,19,31,36,40,17,20,33,39);(1,3,4,8)(2,6,5,7)(11,22)(16,31)(20,34)(23,25)(24,37)(29,38)(32,39)(35,40)8384 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 6255);