Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C131 · Schur multiplier (H2) = C26 |
| direct product | C131×#33371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C131 | C22×#99 |
| commutator | C2×C4 | C23×C4×C131 |
| Frattini, cosocle | C2×Q8 | C24×C131 |
| prime radical | #3500×C131 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C131 ◁ C2×Q8×C131 ◁ C2×H(5,𝔽2)×C131 ◁ G | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 163 |
| transitive degree | μt(G) = 4192 |
A minimal faithful permutation representation — 4 generators on 163 points:
(1,2,7,12,8,13,4,10)(3,14,15,30,17,25,22,32)(5,11,19,29,6,20,18,28)(9,23,26,21,27,31,24,16)(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)(5,16)(6,21)(7,22)(8,17)(10,24)(11,25)(12,26)(13,27)(14,20)(18,23)(19,31)(28,30)(29,32);(1,4,8,7)(2,10,13,12)(3,15,17,22)(5,19,6,18)(9,24,27,26)(11,29,20,28)(14,30,25,32)(16,31,21,23);(1,5,8,6)(2,11,13,20)(3,16,17,21)(4,18,7,19)(9,25,27,14)(10,28,12,29)(15,23,22,31)(24,30,26,32)6026 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 26359);