Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C137 · Schur multiplier (H2) = C25 |
| direct product | C137×#17677 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C137 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C137 |
| Frattini, cosocle | C42 | C24×C137 |
| prime radical | #3444×C137 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C137 ◁ C22×C42×C137 ◁ G | |
976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 4 generators on 169 points:
(1,2,5,9,6,10,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,27,29,32)(20,25)(21,26)(22,24)(23,28,31,30)(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)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(18,24)(23,31)(25,26)(27,32);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,16,11,15)(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,27,31,32)7124 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 10665);