Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#11374 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #127 |
| commutator | C22×C4 | C22×C4×C103 |
| Frattini, cosocle | C23×C4 | C23×C103 |
| socle | C22×C103 | #399 |
| prime radical | C24×C103 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C23×C4×C103 ◁ G | |
598 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)(7,16,20,32)(8,21)(9,14)(10,15)(11,25)(12,18,28,19)(13,24)(17,33,34,29)(22,36,35,27)(23,31)(26,30)(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);(2,4)(5,7,11,18,10,20,9,19)(6,12,25,16,15,28,14,32)(8,22,26,34,24,35,23,17)(13,29,31,27,21,33,30,36);(2,4)(5,8,11,26,10,24,9,23)(6,13,25,31,15,21,14,30)(7,17,18,22,20,34,19,35)(12,27,16,33,28,36,32,29)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 4362);