Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C103 · Schur multiplier (H2) = C25 |
| direct product | C103×#33443 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C103 | C22×D4 |
| commutator | C4 | C23×C8×C103 |
| Frattini, cosocle | C2×C8 | C24×C103 |
| socle | C2×C103 | C2×C8×D4 |
| prime radical | C2×#135×C103 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C103 ◁ C2×M32×C103 ◁ G | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2,6,11,5,12,20,29,7,9,22,25,21,10,26,28)(3,13,15,19,16,24,30,18,17,27,31,32,14,8,23,4)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135);(1,3)(2,8)(4,11)(5,16)(6,15)(7,17)(9,24)(10,27)(12,13)(14,21)(18,25)(19,29)(20,30)(22,31)(23,26)(28,32);(1,4,6,13,5,19,20,24,7,18,22,27,21,32,26,8)(2,3,11,15,12,16,29,30,9,17,25,31,10,14,28,23);(2,9)(4,18)(8,24)(10,12)(11,25)(13,27)(19,32)(28,29)9064 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26431);