Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C137 · Schur multiplier (H2) = C28 |
| direct product | C4×#564×C137 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C137 | C24 |
| commutator | C22 | C24×C4×C137 |
| Frattini, cosocle | C23 | C25×C137 |
| socle, prime radical | C23×C137 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C137 ◁ G | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 173 |
| transitive degree | μt(G) = 35072 |
A minimal faithful permutation representation — 5 generators on 173 points:
(1,2,3,4)(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);(5,6,9,12)(7,13,14,10)(8,16,18,11)(15,19,17,20)(21,22)(23,26)(24,31)(25,28)(27,33)(29,36)(30,35)(32,34);(5,7)(6,10)(8,17)(9,14)(11,20)(12,13)(15,18)(16,19)(21,23,25,30)(22,26,28,35)(24,32,33,29)(27,36,31,34);(7,14)(10,13)(15,17)(19,20)(21,24,25,33)(22,27,28,31)(23,29,30,32)(26,34,35,36);(5,8,9,18)(6,11,12,16)(7,15,14,17)(10,19,13,20)(22,28)(26,35)(27,31)(34,36)10412 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 27343);