Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C103 · Schur multiplier (H2) = C25 |
| direct product | C103×#33329 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C103 | C22×C42 |
| commutator | C4 | C22×C42×C103 |
| Frattini, cosocle | C2×C8 | C24×C103 |
| socle | C2×C103 | C22×#97 |
| prime radical | C2×#41×C103 | C2×C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C103 ◁ G | |
716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2,4,9,5,11,16,24,6,10,15,25,17,7,18,20)(3,8,13,22,14,23,28,32,12,21,26,31,27,19,29,30)(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);(2,7,10,11)(4,15)(8,19,21,23)(9,24,25,20)(13,26)(16,18)(22,32,31,30)(28,29);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,22)(10,21)(11,23)(15,26)(16,28)(17,27)(18,29)(20,30)(24,32)(25,31);(3,12)(8,21)(13,26)(14,27)(19,23)(22,31)(28,29)(30,32)8446 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26317);