Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C137 · Schur multiplier (H2) = C25 |
| direct product | C137×#17386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C137 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C137 |
| Frattini, cosocle | C22×C4 | C24×C137 |
| prime radical | C2×#538×C137 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C137 ◁ C2×C4×D4×C137 ◁ G | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 4384 |
A minimal faithful permutation representation — 4 generators on 169 points:
(1,2,4,9,6,10,15,23)(3,8,13,21,11,22,25,32)(5,7,14,18,16,19,28,30)(12,17,26,29,24,20,27,31)(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);(2,7)(3,11)(4,14)(8,20)(10,19)(12,24)(13,27)(15,28)(17,22)(21,32)(25,26)(29,31);(1,3)(2,8)(4,13)(5,12)(6,11)(7,17)(9,21)(10,22)(14,26)(15,25)(16,24)(18,29)(19,20)(23,32)(27,28)(30,31);(3,12)(8,17)(11,24)(13,26)(20,22)(21,29)(25,27)(31,32)7946 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 10374);