Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C103 · Schur multiplier (H2) = C23 |
| direct product | C103×#13402 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C103 | #2034 |
| commutator, Frattini, cosocle | C22×C8 | C23×C103 |
| prime radical | C2×#472×C103 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C103 ◁ C22×C4×C103 ◁ C2×#131×C103 ◁ G | |
404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 3 generators on 135 points:
(2,8)(3,13)(4,6)(5,19)(9,24)(10,27)(11,12)(14,32)(15,20)(16,30)(17,31)(18,21)(22,25)(26,28)(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);(1,2)(3,14)(4,11)(5,10)(6,8)(7,12)(9,15)(13,28)(16,26)(17,22)(18,29)(19,23)(20,32)(21,27)(24,31)(25,30);(1,3)(2,9,12,26)(4,16)(5,20)(6,15)(7,17)(8,22,11,14)(10,28,27,24)(13,19)(18,31)(21,30)(23,25,29,32)3502 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 6390);