Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C22 |
| direct product | C137×#8842 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C137 | #120 |
| commutator | C22×C4 | C22×C4×C137 |
| Frattini, cosocle | C2×C42 | C23×C137 |
| socle | C22×C137 | #385 |
| prime radical | C2×#41×C137 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C137 ◁ C2×C42×C137 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 3 generators on 161 points:
(1,2,5,3,4,6,8,7)(9,10,15,20)(11,21,12,23)(13,19,14,18)(16,24,17,22)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161);(1,3,5,6,4,7,8,2)(9,11,15,12)(10,16,20,17)(13,24,14,22)(18,23,19,21);(2,6)(3,7)(9,12,15,11)(10,17,20,16)(13,24,14,22)(18,23,19,21)7124 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 1830);