Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C23 |
| direct product | C137×#13200 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C137 | #127 |
| commutator | C22×C4 | C22×C4×C137 |
| Frattini, cosocle | C22×C8 | C23×C137 |
| socle | C2×C137 | #1955 |
| prime radical | #525×C137 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C137 ◁ C22×C8×C137 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 4384 |
A minimal faithful permutation representation — 3 generators on 169 points:
(1,2,6,12,7,11,23,27,8,3,9,15,5,14,22,30)(4,16,20,31,21,13,28,29,18,17,24,32,19,10,25,26)(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);(1,3,6,15,7,14,23,30,8,2,9,12,5,11,22,27)(4,17,24,31,21,10,25,29,18,16,20,32,19,13,28,26);(1,4,5,19,8,18,7,21)(2,10,11,16,3,13,14,17)(6,20,22,25,9,24,23,28)(12,26,27,31,15,29,30,32)7124 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 6188);