Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C131 · Schur multiplier (H2) = C25 |
| direct product | C131×#23362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | #2380 |
| commutator | C2×C4 | C23×C4×C131 |
| Frattini, cosocle | C22×C4 | C24×C131 |
| prime radical | #3153×C131 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C22×C4×C131 ◁ G | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 4192 |
A minimal faithful permutation representation — 4 generators on 163 points:
(1,2)(3,7,13,23)(4,15,18,25)(5,11)(6,10)(8,19,26,28)(9,17)(12,24)(14,22,20,27)(16,21)(29,32)(30,31)(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);(1,3)(2,7)(4,16)(5,13)(6,20)(8,24)(9,22)(10,27)(11,23)(12,28)(14,17)(15,21)(18,30)(19,29)(25,31)(26,32);(1,4,17,28,5,18,6,19)(2,8,10,15,11,26,9,25)(3,12,20,16,13,29,14,30)(7,21,22,24,23,31,27,32);(2,9,11,10)(3,13)(7,22,23,27)(8,25,26,15)(12,29)(14,20)(16,30)(21,24,31,32)5240 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 16350);