Finite Group DatabasebetaFinite Group Databasebeta| order | 25856 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C101 · Schur multiplier (H2) = C24×C4 |
| direct product | C101×#21071 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C101 | C22×D4 |
| commutator | C2×C4 | C23×C4×C101 |
| Frattini, cosocle | C22×C4 | C24×C101 |
| socle | C22×C101 | C2×C4×D4 |
| prime radical | C22×#41×C101 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C101 ◁ C22×M16×C101 ◁ G | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 3232 |
A minimal faithful permutation representation — 4 generators on 121 points:
(2,3)(5,6,9,13,10,8,17,12)(7,11,16,19,15,14,20,18)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(1,2,4,3)(5,7,9,16,10,15,17,20)(6,11,13,19,8,14,12,18);(1,3,4,2)(5,8)(6,10)(7,14)(9,12)(11,15)(13,17)(16,18)(19,20);(7,15)(11,14)(16,20)(18,19)6464 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(25856, 14059);