Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C137 · Schur multiplier (H2) = C25 |
| direct product | C2×#2098×C137 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C137 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C24×C137 |
| socle | C22×C137 | #399 |
| prime radical | C22×#41×C137 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C137 ◁ C23×C4×C137 ◁ G | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 171 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 4 generators on 171 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171);(3,4,7,12,8,13,9,23)(5,14,16,26,18,30,17,10)(6,19,21,29,22,11,24,28)(15,31,20,34,32,27,33,25);(3,5,9,16,8,18,7,17)(4,10,23,14,13,26,12,30)(6,20,24,32,22,33,21,15)(11,27,29,25,19,31,28,34);(3,6)(4,11)(5,15)(7,21)(8,22)(9,24)(10,25)(12,28)(13,19)(14,31)(16,20)(17,33)(18,32)(23,29)(26,34)(27,30)7124 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 16727);