Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C24 |
| direct product | C137×#13011 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C137 | C2×D8 |
| commutator | C2×C8 | C22×C4×C137 |
| Frattini, cosocle | C22×C8 | C23×C137 |
| socle | C22×C137 | C4×D8 |
| prime radical | C2×#508×C137 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C137 ◁ C22×C8×C137 ◁ C2×C4×C8×C137 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 173 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 3 generators on 173 points:
(2,4)(5,6,11,18,12,19,9,16,13,20,29,35,21,30,10,17)(7,15,23,14,24,32,8,26,25,31,28,33,27,36,22,34)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173);(1,2,3,4)(5,7,21,24,13,25,12,27)(6,14,30,26,20,33,19,34)(8,10,28,29,22,9,23,11)(15,16,32,18,31,17,36,35);(5,8)(6,15)(7,10)(9,25)(11,27)(12,23)(13,22)(14,16)(17,33)(18,26)(19,36)(20,31)(21,28)(24,29)(30,32)(34,35)7946 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 5999);