Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C25 |
| direct product | C103×#33411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | C22×D8 |
| commutator | C8 | C23×C4×C103 |
| Frattini, cosocle | C2×C8 | C24×C103 |
| prime radical | C2×#41×C103 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C2×#41×C103 ◁ C2×#131×C103 ◁ G | |
658 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)(6,11)(8,19)(9,20)(12,21)(13,22)(14,26)(15,25)(17,32)(18,31)(23,24)(27,30)(28,33)(29,36)(34,35)(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);(5,6,12,26,9,14,8,11,10,15,19,24,20,23,21,25)(7,13,27,36,18,29,17,22,16,28,32,35,31,34,30,33);(5,7)(6,13)(8,17)(9,18)(10,16)(11,22)(12,27)(14,29)(15,28)(19,32)(20,31)(21,30)(23,34)(24,35)(25,33)(26,36);(7,16)(13,28)(17,30)(18,31)(22,33)(27,32)(29,34)(35,36)6592 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26399);