Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C131 · Schur multiplier (H2) = C23×C4 |
| direct product | C131×#33705 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C131 |
| prime radical | C23×C131 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C2×C42×C131 ◁ C2×C4×C8×C131 ◁ G | |
578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 171 |
| transitive degree | μt(G) = 16768 |
A minimal faithful permutation representation — 4 generators on 171 points:
(1,2)(3,6)(4,5)(7,8)(9,10,16,22)(11,18,23,35)(12,17,26,32)(13,20,29,38)(14,19,30,37)(15,31,28,21)(24,36,39,34)(25,40,27,33)(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)(6,8)(9,11,14,24,15,25,29,26,16,23,30,39,28,27,13,12)(10,17,20,33,21,34,37,35,22,32,38,40,31,36,19,18);(1,3,4,7)(2,6,5,8)(9,12,13,27,28,39,30,23,16,26,29,25,15,24,14,11)(10,18,19,36,31,40,38,32,22,35,37,34,21,33,20,17);(11,23)(12,26)(17,32)(18,35)(24,39)(25,27)(33,40)(34,36)6812 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 26693);