Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C22 |
| direct product | C137×#13184 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C137 | C2×#106 |
| commutator | C2×C8 | C22×C4×C137 |
| Frattini, cosocle | C22×C8 | C23×C137 |
| prime radical | C23×C137 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C137 ◁ C2×C42×C137 ◁ C2×C4×C8×C137 ◁ G | |
294 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)(7,15,17,31)(8,19)(9,12)(10,24)(11,16,26,32)(13,22)(14,23)(18,33,25,29)(20,27,35,30)(21,34)(28,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173);(5,7,10,25,9,17,23,18)(6,11,14,30,12,26,24,27)(8,20,22,16,21,35,36,32)(13,29,19,15,28,33,34,31);(1,2,3,4)(6,12)(11,26)(13,28)(14,24)(15,31)(19,34)(27,30)(29,33)6302 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 6172);