Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C24 |
| direct product | C137×#11310 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C137 | #387 |
| commutator | C22×C4 | C22×C4×C137 |
| Frattini, cosocle | C23×C4 | C23×C137 |
| socle | C2×C137 | #1443 |
| prime radical | C24×C137 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C137 ◁ C23×C4×C137 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 4384 |
A minimal faithful permutation representation — 3 generators on 169 points:
(1,2,6,12,7,10,5,11)(3,13,15,28,16,8,21,23)(4,9,20,27,18,25,19,26)(14,29,17,32,30,22,31,24)(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)(2,8)(4,17,18,31)(5,15)(6,21)(7,16)(9,24,25,32)(10,13)(11,23)(12,28)(14,19,30,20)(22,26,29,27);(1,4)(2,9,10,25)(3,14)(5,20)(6,19)(7,18)(8,22,13,29)(11,27,12,26)(15,31)(16,30)(17,21)(23,32,28,24)6302 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 4298);