Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C25 |
| direct product | C103×#25191 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| prime radical | C103×#3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C22×C4×C103 ◁ G | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 generators on 139 points:
(5,6,10,17,12,18,11,16)(7,19,23,33,24,32,22,13)(8,25,28,15,9,29,27,14)(20,30,21,31,36,35,26,34)(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);(1,2)(3,4)(5,7,10,23,12,24,11,22)(6,13,17,19,18,33,16,32)(8,26,28,20,9,21,27,36)(14,34,25,30,15,31,29,35);(2,4)(5,8,12,9)(6,14,18,15)(7,20,24,36)(10,27,11,28)(13,30,33,35)(16,25,17,29)(19,34,32,31)(21,23,26,22);(5,9,12,8)(6,15,18,14)(7,21,24,26)(10,27,11,28)(13,31,33,34)(16,25,17,29)(19,35,32,30)(20,23,36,22)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 18179);