Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#26505 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| prime radical | C23×C103 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C22×C4×C103 ◁ G | |
668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 generators on 143 points:
(1,2,4,5)(3,7,8,6)(9,10,16,22)(11,23,28,36)(12,29,32,39)(13,33,30,38)(14,21,15,20)(17,27,37,26)(18,25,40,31)(19,24,35,34)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(2,5)(6,7)(9,11,15,27,16,28,14,26)(10,17,21,36,22,37,20,23)(12,30,31,34,32,13,25,24)(18,38,29,35,40,33,39,19);(9,12,15,31,16,32,14,25)(10,18,21,29,22,40,20,39)(11,24,27,30,28,34,26,13)(17,35,36,33,37,19,23,38);(1,3,4,8)(2,6,5,7)(9,13,15,24,16,30,14,34)(10,19,21,38,22,35,20,33)(11,25,27,12,28,31,26,32)(17,18,36,29,37,40,23,39)4120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 19493);