Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C137 · Schur multiplier (H2) = C24 |
| direct product | C137×#20324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C137 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C137 |
| prime radical | #555×C137 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C137 ◁ C42×C137 ◁ G | |
570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 17536 |
A minimal faithful permutation representation — 4 generators on 169 points:
(1,2,6,4)(3,9,12,16)(5,14,15,8)(7,10,13,11)(17,18,22,25)(19,23,21,30)(20,27,29,31)(24,28,32,26)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169);(1,3,6,12)(2,7,4,13)(5,10,15,11)(8,16,14,9)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(1,4,6,2)(3,7,12,13)(5,14,15,8)(9,10,16,11)(17,20,22,29)(18,24,25,32)(19,26,21,28)(23,31,30,27);(3,10)(5,15)(7,16)(8,14)(9,13)(11,12)(18,25)(23,30)(24,32)(27,31)5069 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 13312);