Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C137 · Schur multiplier (H2) = C24 |
| direct product | C137×#17953 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C137 | C22×D4 |
| commutator | C2×C4 | C23×C4×C137 |
| Frattini, cosocle | C42 | C24×C137 |
| socle | C22×C137 | C2×C4×D4 |
| prime radical | C23×C137 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C137 ◁ #431×C137 ◁ G | |
558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 17536 |
A minimal faithful permutation representation — 4 generators on 169 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,22,25,23,19,21,27)(20,24,30,31,28,26,29,32)(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);(3,9)(6,12)(10,15)(13,16)(17,19,23,18)(20,24,28,26)(21,25,22,27)(29,32,30,31);(1,3,4,10,5,9,11,15)(2,6,8,13,7,12,14,16)(17,20,21,29,23,28,22,30)(18,24,27,32,19,26,25,31);(2,7)(6,12)(8,14)(13,16)(20,28)(24,26)(29,30)(31,32)7124 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 10941);