Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C25 |
| direct product | C103×#25438 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| prime radical | #535×C103 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C22×C4×C103 ◁ G | |
892 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)(7,18)(8,13)(9,24)(10,16)(11,17)(12,20)(14,25)(15,22)(19,33)(21,28)(23,27)(26,31)(29,35)(30,34)(32,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(1,3,4,2)(5,7,10,20,11,8,22,21)(6,12,15,18,17,28,16,13)(9,19,25,34,26,23,36,35)(14,27,24,30,32,33,31,29);(2,3)(5,8)(6,13)(7,11)(9,19)(10,21)(12,15)(14,30)(16,28)(17,18)(20,22)(23,26)(24,33)(25,34)(27,31)(29,32)(35,36);(5,9,22,36,11,26,10,25)(6,14,16,31,17,32,15,24)(7,19,21,35,8,23,20,34)(12,27,13,29,28,33,18,30)4120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 18426);