Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C131 · Schur multiplier (H2) = C24×C4 |
| direct product | C131×#21072 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C131 | C22×D4 |
| commutator | C2×C4 | C23×C4×C131 |
| Frattini, cosocle | C22×C4 | C24×C131 |
| socle | C22×C131 | C2×C4×D4 |
| prime radical | C22×#41×C131 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C131 ◁ C22×M16×C131 ◁ G | |
1018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 4 generators on 167 points:
(2,3)(5,6,10,15,11,14,9,23)(7,12,19,26,17,27,18,33)(8,20,21,13,22,29,24,28)(16,30,32,25,31,35,34,36)(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);(1,2,4,3)(5,7,10,19,11,17,9,18)(6,12,15,26,14,27,23,33)(8,16,21,32,22,31,24,34)(13,25,29,35,28,36,20,30);(1,3,4,2)(5,8)(6,13)(7,16)(9,24)(10,21)(11,22)(12,25)(14,28)(15,29)(17,31)(18,34)(19,32)(20,23)(26,35)(27,36)(30,33);(6,14)(7,17)(13,28)(15,23)(16,31)(18,19)(20,29)(32,34)8384 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 14060);