Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C137 · Schur multiplier (H2) = C26 |
| direct product | C137×#33550 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C137 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C24×C137 |
| socle | C2×C137 | C2×B(4,𝔽2) |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C137 ◁ C2×C4×C137 ◁ C2×#41×C137 ◁ G | |
1144 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)(3,11)(4,8)(5,9)(6,10)(7,14)(12,23)(13,24)(15,20)(16,21)(17,22)(18,26)(19,27)(25,31)(28,30)(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);(1,3)(2,7)(4,13)(5,14)(6,12)(8,19)(9,11)(10,18)(15,27)(16,25)(17,26)(20,24)(21,29)(22,23)(28,32)(30,31);(3,12)(5,17)(7,18)(11,23)(13,25)(15,28)(19,29)(24,31);(1,4,6,16)(2,8,10,21)(3,13,12,25)(5,15,17,28)(7,19,18,29)(9,20,22,30)(11,24,23,31)(14,27,26,32)4658 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 26538);