Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C131 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#2249×C131 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C131 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C131 |
| prime radical | C22×D16×C131 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C131 ◁ C2×C42×C131 ◁ C2×C4×C8×C131 ◁ G | |
854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 16768 |
A minimal faithful permutation representation — 4 generators on 169 points:
(1,2)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169);(4,6)(7,8)(9,17)(10,22)(11,14)(12,25)(13,19)(15,18)(16,23)(20,29)(21,36)(24,37)(26,35)(27,38)(28,33)(30,32)(31,34);(7,9,18,32,12,21,24,35,11,20,33,38,23,34,10,19)(8,13,22,31,16,27,28,29,14,26,37,36,25,30,15,17);(3,4,5,6)(8,14)(13,26)(15,28)(16,25)(17,29)(22,37)(27,30)(31,36)9956 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 26700);