Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C131 · Schur multiplier (H2) = C23 |
| direct product | C131×#13374 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | #2023 |
| commutator, Frattini, cosocle | C22×C8 | C23×C131 |
| prime radical | C2×#135×C131 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C22×C4×C131 ◁ #405×C131 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 4192 |
A minimal faithful permutation representation — 3 generators on 163 points:
(1,2)(3,11,13,28)(4,16)(5,20)(6,9)(7,14,12,30)(8,18)(10,22)(15,24,21,29)(17,26,32,23)(19,27)(25,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163);(1,2)(3,12,13,7)(4,8)(5,10)(6,9)(11,14,28,30)(15,26,21,23)(16,18)(17,29,32,24)(19,25)(20,22)(27,31);(1,3)(2,7)(4,17)(5,21)(6,13)(8,24)(9,12)(10,26)(11,27)(14,31)(15,22)(16,28)(18,30)(19,32)(20,23)(25,29)4454 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 6362);