Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C137 · Schur multiplier (H2) = C25 |
| direct product | C137×#22695 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C137 | D42 |
| commutator | C2×C4 | C23×C4×C137 |
| Frattini, cosocle | C22×C4 | C24×C137 |
| socle | C2×C137 | D4×C4⋊C4 |
| prime radical | C2×#135×C137 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C137 ◁ C22×C4×C137 ◁ G | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 4384 |
A minimal faithful permutation representation — 4 generators on 169 points:
(1,2,6,13,8,14,7,12)(3,15,19,27,20,31,18,9)(4,21,22,29,23,26,17,10)(5,11,25,30,16,28,24,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169);(1,3)(2,9)(4,5)(6,19)(7,18)(8,20)(10,28)(11,29)(12,31)(13,15)(14,27)(16,23)(17,24)(21,32)(22,25)(26,30);(1,4,7,17,8,23,6,22)(2,10,13,21,14,29,12,26)(3,16,18,25,20,5,19,24)(9,11,15,30,27,28,31,32);(1,5)(2,11)(3,17)(4,19)(6,25)(7,24)(8,16)(9,26)(10,15)(12,32)(13,30)(14,28)(18,23)(20,22)(21,27)(29,31)7124 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 15683);