Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C25 |
| direct product | C103×#25433 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| prime radical | C2×#472×C103 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C22×C4×C103 ◁ G | |
916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 generators on 139 points:
(1,2,4,3)(5,6,10,16)(7,12,20,28)(8,21,24,30)(9,15,22,14)(11,19,26,18)(13,29,31,23)(17,32,35,25)(27,34,36,33)(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);(1,3,4,2)(5,7,9,18,10,20,22,19)(6,11,14,12,16,26,15,28)(8,17,23,33,24,35,29,34)(13,25,21,27,31,32,30,36);(2,3)(5,7)(6,12)(8,17)(9,18)(10,20)(11,15)(13,27)(14,26)(16,28)(19,22)(21,32)(23,33)(24,35)(25,30)(29,34)(31,36);(5,8,22,29,10,24,9,23)(6,13,15,30,16,31,14,21)(7,17,19,34,20,35,18,33)(11,25,28,36,26,32,12,27)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 18421);