Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C137 · Schur multiplier (H2) = C26 |
| direct product | C137×#10773 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C137 | #121 |
| commutator | C2×C4 | C2×C42×C137 |
| Frattini, cosocle | C23×C4 | C23×C137 |
| socle | C22×C137 | #380 |
| prime radical | C22×#41×C137 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C137 ◁ C23×C4×C137 ◁ G | |
1186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 173 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 3 generators on 173 points:
(1,2)(3,4)(5,6)(7,17)(8,22,25,30)(9,15)(10,14)(11,16)(12,19)(13,24,29,23)(18,32,35,26)(20,28)(21,27)(31,34,36,33)(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);(2,4)(5,7,10,20,11,21,9,19)(6,12,15,27,16,28,14,17)(8,18,24,34,25,35,23,33)(13,26,30,36,29,32,22,31);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,22)(15,30)(16,29)(17,31)(19,33)(20,34)(21,35)(27,36)(28,32)8768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 3761);