Finite Group DatabasebetaFinite Group Databasebeta| order | 33536 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C131 · Schur multiplier (H2) = C23 |
| direct product | C131×#13384 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C131 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C131 |
| prime radical | #3061×C131 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C131 ◁ C22×C4×C131 ◁ C2×#131×C131 ◁ G | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 167 |
| transitive degree | μt(G) = 16768 |
A minimal faithful permutation representation — 3 generators on 167 points:
(2,3)(5,6)(7,11)(8,12)(9,14)(10,16)(13,21)(15,22)(17,30)(18,27)(19,25)(20,26)(23,29)(24,31)(28,34)(32,36)(33,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167);(1,2)(3,4)(5,7)(6,11)(8,18)(9,17)(10,20)(12,27)(13,24)(14,30)(15,25)(16,26)(19,22)(21,31)(23,32)(28,35)(29,36)(33,34);(1,3,4,2)(6,12)(7,17)(8,21)(9,22)(11,24)(13,15)(14,28)(16,29)(18,32)(19,20)(23,34)(25,36)(26,35)(27,30)5240 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33536, 6372);