Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C25 |
| direct product | C103×#33613 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C103 |
| prime radical | C103×#3061 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C2×#41×C103 ◁ C2×#131×C103 ◁ G | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 4 generators on 139 points:
(1,2)(3,4)(5,6,10,15)(7,17,20,33)(8,23,24,13)(9,27,29,34)(11,30,28,16)(12,31,26,21)(14,32,19,18)(22,36,35,25)(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);(5,7,18,23,11,22,29,31,10,20,32,13,28,35,9,21)(6,12,27,25,16,8,19,33,15,26,34,36,30,24,14,17);(2,4)(5,8)(6,13)(7,19)(9,25)(10,24)(11,26)(12,28)(14,20)(15,23)(16,21)(17,32)(18,33)(22,34)(27,35)(29,36)(30,31);(7,20)(8,24)(12,26)(13,23)(17,33)(21,31)(22,35)(25,36)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26601);