Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C131 · Schur multiplier (H2) = C25 |
| direct product | C131×#23197 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C131 | D42 |
| commutator | C2×C4 | C23×C4×C131 |
| Frattini, cosocle | C22×C4 | C24×C131 |
| socle | C2×C131 | #2366 |
| prime radical | #3150×C131 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C131 ◁ C22×C4×C131 ◁ G | |
880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 4192 |
A minimal faithful permutation representation — 4 generators on 163 points:
(1,2)(3,13,18,29)(4,19)(5,9)(6,10)(7,12)(8,16,26,17)(11,23)(14,31,15,24)(20,27)(21,30,32,25)(22,28)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(1,3,6,17,7,18,23,16)(2,8,10,29,12,26,11,13)(4,14,5,21,20,15,22,32)(9,24,27,25,28,31,19,30);(1,4,6,5,7,20,23,22)(2,9,11,19,12,28,10,27)(3,14,17,21,18,15,16,32)(8,24,13,30,26,31,29,25);(1,5,6,20,7,22,23,4)(2,9,10,27,12,28,11,19)(3,15,16,21,18,14,17,32)(8,25,13,24,26,30,29,31)6812 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 16185);