Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C24 |
| direct product | C137×#13036 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C137 | #1879 |
| commutator | C2×C8 | C22×C4×C137 |
| Frattini, cosocle | C22×C8 | C23×C137 |
| prime radical | C2×#509×C137 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C137 ◁ C22×C4×C137 ◁ #405×C137 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 4384 |
A minimal faithful permutation representation — 3 generators on 169 points:
(1,2,7,12,14,17,6,11,9,13,25,26,8,3,5,16)(4,18,21,10,27,28,19,15,23,31,24,30,22,32,20,29)(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);(1,3,14,13,9,17,8,2)(4,15,22,30,23,29,27,10)(5,16,7,26,6,11,25,12)(18,21,28,20,31,24,32,19);(1,4)(2,10)(3,15)(5,24)(6,21)(7,19)(8,27)(9,23)(11,28)(12,31)(13,30)(14,22)(16,32)(17,29)(18,26)(20,25)5480 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 6024);