Finite Group DatabasebetaFinite Group Databasebeta| order | 35072 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C137 · Schur multiplier (H2) = C2×C4 |
| direct product | C137×#9859 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C137 | #127 |
| commutator | C42 | C22×C4×C137 |
| Frattini, cosocle | C2×C42 | C23×C137 |
| socle | C22×C137 | #390 |
| prime radical | C23×C137 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C137 ◁ C2×C42×C137 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 169 |
| transitive degree | μt(G) = 17536 |
A minimal faithful permutation representation — 3 generators on 169 points:
(1,2)(3,9)(4,12)(5,7)(6,11)(8,13)(10,15)(14,16)(17,18,21,25,22,26,20,23)(19,27,30,24,28,31,29,32)(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);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(18,23,26,25)(19,28)(24,31,32,27)(29,30);(2,7)(3,10)(8,12)(11,16)(17,19,20,29,22,28,21,30)(18,24,25,31,26,32,23,27)7124 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35072, 2847);