Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C137 · Schur multiplier (H2) = C4 |
| direct product | C137×#7532 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C137 | #1408 |
| commutator | C4×C8 | C2×C4×C137 |
| Frattini, cosocle | #494 | C22×C137 |
| prime radical | C22×C137 | #353 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C137 ◁ C22×C137 ◁ C2×C4×C137 ◁ C42×C137 ◁ #494×C137 ◁ G | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 4384 |
A minimal faithful permutation representation — 2 generators on 169 points:
(1,2,5,11,8,13,3,14)(4,10,19,26,22,29,18,9)(6,12,16,27,7,24,17,30)(15,31,20,28,23,25,21,32)(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);(1,3,8,5)(2,9,13,26)(4,18,22,19)(6,17,7,16)(10,27,29,30)(11,28,14,32)(12,31,24,25)(15,21,23,20)3425 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 520);