Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#23833 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C103 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| socle | C2×C103 | #2398 |
| prime radical | #575×C103 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C103 ◁ C22×C8×C103 ◁ G | |
616 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)(3,12,15,25)(4,9)(5,10)(6,20)(7,11)(8,16,22,13)(14,26,28,32)(17,23)(18,30)(19,24)(21,29,31,27)(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);(3,13,15,16)(8,12,22,25)(14,27,28,29)(21,26,31,32);(1,3)(2,8)(4,14)(5,13)(6,16)(7,15)(9,21)(10,25)(11,22)(12,20)(17,27)(18,29)(19,28)(23,32)(24,31)(26,30);(1,4,5,17,7,19,6,18)(2,9,10,23,11,24,20,30)(3,14,13,27,15,28,16,29)(8,21,25,32,22,31,12,26)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 16821);