Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C131 · Schur multiplier (H2) = C2×C4 |
| direct product | C131×#11963 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | C2×#102 |
| commutator | C2×C8 | C22×C4×C131 |
| Frattini, cosocle | C4×C8 | C23×C131 |
| prime radical | #3147×C131 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C2×C42×C131 ◁ C2×C4×C8×C131 ◁ G | |
396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 171 |
| transitive degree | μt(G) = 16768 |
A minimal faithful permutation representation — 3 generators on 171 points:
(1,2,3,6,4,5,7,8)(9,10,14,19,12,21,28,18,15,20,31,36,29,17,13,30)(11,22,25,34,27,35,40,38,26,37,39,16,23,33,24,32)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171);(2,5)(3,7)(9,11)(10,16)(12,23)(13,25)(14,24)(15,26)(17,32)(18,35)(19,37)(20,34)(21,38)(22,36)(27,29)(28,39)(30,33)(31,40);(2,5)(6,8)(9,12,15,29)(10,17,20,21)(11,23,26,27)(13,14,28,31)(16,32,34,38)(18,19,30,36)(22,35,37,33)(24,39,40,25)6812 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 4951);