Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C131 · Schur multiplier (H2) = C25 |
| direct product | C2×#2170×C131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C131 | D4 |
| commutator | C4 | C23×C8×C131 |
| Frattini, cosocle | C2×C8 | C24×C131 |
| socle | C22×C131 | C8×D4 |
| prime radical | C2×#135×C131 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C131 ◁ C22×C16×C131 ◁ G | |
662 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);(3,4,8,12,7,13,22,30,9,14,24,31,23,11,27,29)(5,15,17,20,18,26,32,21,19,28,33,34,16,10,25,6);(3,5)(4,10)(6,12)(7,18)(8,17)(9,19)(11,28)(13,15)(14,26)(16,23)(20,30)(21,31)(22,32)(24,33)(25,27)(29,34);(3,6,8,15,7,20,22,26,9,21,24,28,23,34,27,10)(4,5,12,17,13,18,30,32,14,19,31,33,11,16,29,25)14672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 26427);