Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C137 · Schur multiplier (H2) = C2×C4 |
| direct product | C137×#12668 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C137 | #106 |
| commutator | C8 | C2×C42×C137 |
| Frattini, cosocle | C22×C8 | C23×C137 |
| socle | C2×C137 | C4×#106 |
| prime radical | C23×C137 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C137 ◁ C22×C8×C137 ◁ C2×C4×C8×C137 ◁ G | |
308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 4384 |
A minimal faithful permutation representation — 3 generators on 169 points:
(1,2,6,12,7,14,20,27,8,13,5,11,19,26,4,10)(3,15,16,25,17,28,21,24,18,31,32,30,23,29,22,9)(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)(2,9,13,24)(4,16)(5,21)(6,22)(7,23)(8,18)(10,15,27,31)(11,28,12,29)(14,30,26,25)(17,19)(20,32);(1,4,19,5,8,20,7,6)(2,10,26,11,13,27,14,12)(3,16,17,21,18,32,23,22)(9,15,25,28,24,31,30,29)8768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 5656);