Finite Group DatabasebetaFinite Group Databasebeta| order | 26001 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C107 · Schur multiplier (H2) = C1 |
| direct product | C107×#6875 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C107 | C9⋊C9 |
| commutator | C9 | C3×C9×C107 |
| Frattini, cosocle | C3×C9 | C32×C107 |
| prime radical | C32×C107 | 3−1+2 |
| LCS | G ▷ C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C107 ◁ C3×C9×C107 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 134 |
| transitive degree | μt(G) = 2889 |
A minimal faithful permutation representation — 2 generators on 134 points:
(2,5,11,7,14,23,18,26,13)(3,8,19)(6,15,12,16,27,24,17,22,25)(9,21,20)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134);(1,2,6,3,5,12,9,13,15,4,7,17,8,14,25,20,11,22,10,18,16,19,26,24,21,23,27)3745 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26001, 22);