Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C137 · Schur multiplier (H2) = C26 |
| direct product | C137×#22459 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C137 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C137 |
| Frattini, cosocle | C22×C4 | C24×C137 |
| prime radical | C2×#577×C137 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C137 ◁ C22×C4×C137 ◁ G | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 4384 |
A minimal faithful permutation representation — 4 generators on 169 points:
(1,2,6,11,7,12,5,20)(3,13,16,8,17,26,21,25)(4,10,18,27,19,9,22,28)(14,30,15,23,32,24,31,29)(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);(1,3,7,17)(2,8,12,25)(4,14,19,32)(5,21,6,16)(9,23,10,29)(11,26,20,13)(15,22,31,18)(24,27,30,28);(1,4)(2,9)(3,14)(5,18)(6,22)(7,19)(8,23)(10,12)(11,27)(13,24)(15,21)(16,31)(17,32)(20,28)(25,29)(26,30);(1,4)(2,10)(3,15)(5,22)(6,18)(7,19)(8,24)(9,12)(11,27)(13,23)(14,21)(16,32)(17,31)(20,28)(25,30)(26,29)7124 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 15447);