Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C131 · Schur multiplier (H2) = C26 |
| direct product | C131×#10741 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | #386 |
| commutator | C2×C4 | C2×C42×C131 |
| Frattini, cosocle | C23×C4 | C23×C131 |
| prime radical | C24×C131 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C23×C4×C131 ◁ G | |
1186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 3 generators on 167 points:
(1,2,3,4)(5,6)(7,15)(8,12)(9,21)(10,14)(11,17)(13,23)(16,31)(18,30)(19,28)(20,27)(22,26)(24,32)(25,35)(29,33)(34,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(5,7)(6,11,14,25)(8,16)(9,17)(10,18)(12,24,27,36)(13,15,21,30)(19,32)(20,33)(22,34)(23,35)(26,31,28,29);(5,8,10,20)(6,12,14,27)(7,16,18,33)(9,22,23,19)(11,24,25,36)(13,28,21,26)(15,29,30,31)(17,34,35,32)7598 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 3729);