Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C131 · Schur multiplier (H2) = C25 |
| direct product | C2×#2221×C131 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C131 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C22×C4×C131 ◁ C2×#131×C131 ◁ G | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 165 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 4 generators on 165 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165);(4,10)(5,16)(6,11)(7,8)(12,27)(13,24)(14,23)(15,25)(17,31)(18,21)(19,20)(22,28)(29,34)(30,33);(3,4,11,24,8,14,20,25,9,15,19,23,7,13,6,10)(5,12,26,27,16,28,31,33,18,29,32,34,21,30,17,22);(3,5)(4,12)(6,17)(7,21)(8,16)(9,18)(10,22)(11,26)(13,30)(14,28)(15,29)(19,32)(20,31)(23,34)(24,27)(25,33)6812 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 26602);