Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C131 · Schur multiplier (H2) = C26 |
| direct product | C131×#33572 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24×C131 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C2×C4×C131 ◁ C2×#41×C131 ◁ G | |
1108 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)(3,14)(4,12)(5,11)(6,21)(7,23)(8,13)(9,15)(10,19)(16,24)(17,31)(18,29)(20,28)(22,27)(25,30)(26,32)(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)(2,9)(4,15)(5,17)(6,22)(7,16)(8,18)(10,24)(11,26)(12,29)(13,27)(14,21)(19,32)(20,30)(23,25)(28,31);(1,4,8,6)(2,10,13,28)(3,15,18,22)(5,19,7,20)(9,24,27,31)(11,21,23,12)(14,25,29,26)(16,30,17,32);(1,5,8,7)(2,11,13,23)(3,16,18,17)(4,19,6,20)(9,25,27,26)(10,21,28,12)(14,24,29,31)(15,30,22,32)4454 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 26560);