Finite Group DatabasebetaFinite Group Databasebeta| order | 26001 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C107 · Schur multiplier (H2) = C33 |
| direct product | C3×C9⋊C9×C107 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C33×C107 | C32 |
| commutator | C3 | C32×C9×C107 |
| Frattini, cosocle | C32 | C33×C107 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33×C107 ◁ G | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 26001 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,3)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128);(5,7,10)(8,12,11)(13,14,16,15,17,19,18,20,21);(4,5,8,6,7,11,9,10,12)10593 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26001, 33);