Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C137 · Schur multiplier (H2) = C25 |
| direct product | C137×#21163 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C137 | C22×D4 |
| commutator | C2×C4 | C23×C4×C137 |
| Frattini, cosocle | C22×C4 | C24×C137 |
| socle | C22×C137 | C2×C4×D4 |
| prime radical | C24×C137 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C137 ◁ C22×M16×C137 ◁ G | |
882 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)(5,6,10,14,11,15,9,24)(7,12,19,27,17,26,18,33)(8,20,22,28,23,34,21,13)(16,29,32,35,30,36,31,25)(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)(5,7,10,19,11,17,9,18)(6,12,14,27,15,26,24,33)(8,16,22,32,23,30,21,31)(13,25,20,29,28,35,34,36);(5,8,9,21,11,23,10,22)(6,13,14,20,15,28,24,34)(7,16,18,31,17,30,19,32)(12,25,27,29,26,35,33,36);(7,17)(12,26)(16,30)(18,19)(25,35)(27,33)(29,36)(31,32)8768 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 14151);