Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C137 · Schur multiplier (H2) = C26 |
| direct product | C137×#10778 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C137 | #379 |
| commutator | C2×C4 | C2×C42×C137 |
| Frattini, cosocle | C23×C4 | C23×C137 |
| socle | C2×C137 | C22⋊#95 |
| prime radical | C22×#41×C137 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C137 ◁ C23×C4×C137 ◁ G | |
1164 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)(3,13,16,8)(4,18,20,26)(5,11)(6,12)(7,10)(9,19,25,21)(14,28)(15,24,17,23)(22,29)(27,31)(30,32)(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);(1,3,6,17,7,16,5,15)(2,8,12,24,10,13,11,23)(4,14,19,29,20,30,21,31)(9,22,18,32,25,27,26,28);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,22)(10,25)(11,26)(12,18)(13,27)(15,31)(16,30)(17,29)(23,28)(24,32)7946 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 3766);