Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C137 · Schur multiplier (H2) = C23×C4 |
| direct product | C137×#10900 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C137 | #120 |
| commutator | C2×C4 | C2×C42×C137 |
| Frattini, cosocle | C23×C4 | C23×C137 |
| socle | C22×C137 | #386 |
| prime radical | C24×C137 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C137 ◁ C23×C4×C137 ◁ G | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 173 |
| transitive degree | μt(G) = 8768 |
A minimal faithful permutation representation — 3 generators on 173 points:
(1,2,4,3)(5,6)(7,16)(8,21)(9,25)(10,14)(11,15)(12,18)(13,23)(17,32)(19,28)(20,27)(22,30)(24,29)(26,34)(31,33)(35,36)(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);(5,7,10,18,11,20,9,19)(6,12,14,27,15,28,25,16)(8,17,23,33,24,34,22,35)(13,26,29,36,30,32,21,31);(1,3,4,2)(5,8)(6,13)(7,17)(9,22)(10,23)(11,24)(12,26)(14,29)(15,30)(16,31)(18,33)(19,35)(20,34)(21,25)(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, 3888);