Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C131 · Schur multiplier (H2) = C26 |
| direct product | C131×#33372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | C22×#99 |
| commutator | C2×C4 | C23×C4×C131 |
| Frattini, cosocle | C2×Q8 | C24×C131 |
| prime radical | #3499×C131 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C2×D4×C131 ◁ C2×2−1+4×C131 ◁ G | |
1176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 4192 |
A minimal faithful permutation representation — 4 generators on 163 points:
(1,2,6,11,7,12,3,13)(4,9,18,25,16,23,14,27)(5,10,15,22,20,26,19,8)(17,24,28,31,29,32,30,21)(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);(1,3,7,6)(2,8,12,22)(4,14,16,18)(5,19,20,15)(9,21,23,31)(10,11,26,13)(17,30,29,28)(24,25,32,27);(1,4)(2,9)(3,14)(5,17)(6,18)(7,16)(8,21)(10,24)(11,25)(12,23)(13,27)(15,28)(19,30)(20,29)(22,31)(26,32);(4,16)(9,23)(14,18)(17,29)(21,31)(24,32)(25,27)(28,30)6026 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 26360);