Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C103 · Schur multiplier (H2) = C26 |
| direct product | C103×#10748 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #120 |
| commutator | C23 | C2×C42×C103 |
| Frattini, cosocle | C23×C4 | C23×C103 |
| socle | C22×C103 | #385 |
| prime radical | C22×#41×C103 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C23×C4×C103 ◁ G | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 139 points:
(1,2)(3,4)(5,6,9,15,11,14,10,16)(7,17,20,30,19,12,24,26)(8,13,22,27,21,29,23,28)(18,25,33,35,32,31,34,36)(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);(5,7,10,20,11,19,9,24)(6,12,16,26,14,17,15,30)(8,18,23,33,21,32,22,34)(13,25,28,35,29,31,27,36);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,21)(12,25)(14,29)(15,27)(16,28)(17,31)(19,32)(20,33)(24,34)(26,35)(30,36)6592 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 3736);