Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C131 · Schur multiplier (H2) = C24×C4 |
| direct product | C131×#25927 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C131 |
| prime radical | C2×#41×C131 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C2×C42×C131 ◁ G | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 171 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 4 generators on 171 points:
(1,2,5,8)(3,7,4,6)(9,10,13,18)(11,20,22,16)(12,24,26,36)(14,28,15,19)(17,30,35,27)(21,38,39,34)(23,32,29,33)(25,31,40,37)(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);(1,3,5,4)(2,6,8,7)(9,11,13,22)(10,16,18,20)(12,25,26,40)(14,23,15,29)(17,34,35,38)(19,32,28,33)(21,30,39,27)(24,37,36,31);(1,3,5,4)(2,7,8,6)(9,12,15,30,13,26,14,27)(10,17,28,36,18,35,19,24)(11,21,29,25,22,39,23,40)(16,31,33,38,20,37,32,34);(1,4,5,3)(2,7,8,6)(10,18)(12,26)(16,20)(19,28)(21,39)(25,40)(27,30)(32,33)5240 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 18915);