Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#29271 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C103 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| socle | C2×C103 | #2580 |
| prime radical | C2×#135×C103 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C103 ◁ C22×C4×C103 ◁ G | |
604 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)(4,16)(5,20)(6,11)(7,14)(8,15)(9,18)(10,21)(13,27)(17,22)(19,30)(23,29)(24,31)(25,28)(26,32)(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)(2,7,11,23)(4,13)(5,14)(6,15)(8,10,12,20)(9,22,26,27)(16,25,30,24)(17,19)(18,28)(21,29)(31,32);(1,3)(2,8,11,12)(4,17)(5,14)(6,15)(7,20,23,10)(9,25,26,24)(13,19)(16,27,30,22)(18,31)(21,29)(28,32);(1,4,5,18,6,19,21,32)(2,9,10,16,11,26,20,30)(3,13,14,28,15,17,29,31)(7,22,12,25,23,27,8,24)4120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 22259);