Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#11350 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #124 |
| commutator | C22×C4 | C22×C4×C103 |
| Frattini, cosocle | C23×C4 | C23×C103 |
| socle | C22×C103 | #397 |
| prime radical | C24×C103 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C23×C4×C103 ◁ G | |
594 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)(7,15)(8,19)(9,22)(10,14)(11,17)(12,21)(13,24)(16,32)(18,31)(20,28)(23,29)(25,33)(26,36)(27,35)(30,34)(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);(5,7,9,17,10,18,24,36)(6,11,13,15,14,26,22,31)(8,16,23,35,21,34,20,33)(12,27,29,32,19,25,28,30);(5,8,9,23,10,21,24,20)(6,12,13,29,14,19,22,28)(7,16,17,35,18,34,36,33)(11,25,15,30,26,27,31,32)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 4338);