Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C131 · Schur multiplier (H2) = C2×C4 |
| direct product | C131×#13139 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | #481 |
| commutator | C2×C8 | C22×C4×C131 |
| Frattini, cosocle | C22×C8 | C23×C131 |
| prime radical | C23×C131 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C22×C4×C131 ◁ C2×#97×C131 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 8384 |
A minimal faithful permutation representation — 3 generators on 167 points:
(2,3)(5,6)(7,15,18,31)(8,11)(9,21)(10,24)(12,17,26,33)(13,23)(14,16)(19,34,25,29)(20,35)(22,28,36,30)(27,32)(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);(1,2,4,3)(6,11)(9,20)(12,26)(14,24)(17,33)(22,36)(23,32)(28,30);(1,3,4,2)(5,7,16,19,8,18,10,25)(6,12,24,28,11,26,14,30)(9,22,32,33,20,36,23,17)(13,29,35,31,27,34,21,15)5240 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 6127);