Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C131 · Schur multiplier (H2) = C25 |
| direct product | C131×#23200 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | D42 |
| commutator | C2×C4 | C23×C4×C131 |
| Frattini, cosocle | C22×C4 | C24×C131 |
| prime radical | C23×C131 | H(5,𝔽2) |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C22×C4×C131 ◁ G | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 4 generators on 167 points:
(1,2,4,3)(5,6,10,15,12,16,11,26)(7,17,19,30,20,29,25,13)(8,21,22,31,23,34,9,14)(18,27,24,32,33,36,35,28)(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);(1,3,4,2)(5,7,12,20)(6,13,16,30)(8,18,23,33)(9,24,22,35)(10,25,11,19)(14,27,31,36)(15,29,26,17)(21,28,34,32);(5,8,10,22,12,23,11,9)(6,14,26,34,16,31,15,21)(7,18,25,35,20,33,19,24)(13,27,17,32,30,36,29,28);(5,9,11,23,12,22,10,8)(6,14,26,34,16,31,15,21)(7,18,25,35,20,33,19,24)(13,28,29,36,30,32,17,27)7598 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 16188);