Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C131 · Schur multiplier (H2) = C25 |
| direct product | C131×#23196 |
| 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 | #3147×C131 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C131 ◁ C22×C4×C131 ◁ G | |
884 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)(3,14,18,27)(4,19)(5,11)(6,12)(7,24)(8,13)(9,17,28,23)(10,20)(15,25,16,29)(21,30)(22,32,31,26)(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);(1,3,6,23,8,18,7,17)(2,9,12,27,13,28,24,14)(4,15,5,22,21,16,20,31)(10,25,19,32,11,29,30,26);(1,4,7,20,8,21,6,5)(2,10,12,19,13,11,24,30)(3,15,17,31,18,16,23,22)(9,25,27,32,28,29,14,26);(1,5,6,21,8,20,7,4)(2,11,12,30,13,10,24,19)(3,16,17,22,18,15,23,31)(9,26,14,29,28,32,27,25)7598 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 16184);