Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#33805 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C103 |
| prime radical | #466×C103 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C2×C42×C103 ◁ C2×C4×C8×C103 ◁ G | |
496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,29)(24,28)(25,27)(26,30)(31,32)(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);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(18,22)(23,30)(24,31)(25,29);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,23)(20,28)(21,27)(22,30)(24,25)(26,32)(29,31);(2,8)(6,9)(7,16)(14,15)(18,22)(23,30)(24,31)(25,29)4120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26793);