Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C137 · Schur multiplier (H2) = C26 |
| direct product | C137×#33521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C137 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C137 |
| prime radical | C22×#135×C137 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C137 ◁ C23×C4×C137 ◁ C23×C8×C137 ◁ G | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 173 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 4 generators on 173 points:
(1,2)(3,4)(5,6,11,16)(7,17,20,34)(8,15,10,13)(9,22,25,30)(12,27,29,19)(14,21,32,24)(18,35,26,31)(23,33,36,28)(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)(6,12)(8,18)(9,23)(10,26)(11,20)(13,28)(14,31)(15,33)(16,29)(17,22)(19,24)(21,27)(25,36)(30,34)(32,35);(2,4);(5,8,11,10)(6,13,16,15)(7,18,20,26)(9,21,25,24)(12,28,29,33)(14,30,32,22)(17,31,34,35)(19,23,27,36)6302 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 26509);