Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C103 · Schur multiplier (H2) = C25 |
| direct product | C103×#33401 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C103 | C22×D4 |
| commutator | C4 | C23×C8×C103 |
| Frattini, cosocle | C2×C8 | C24×C103 |
| socle | C2×C103 | C22×C4⋊C8 |
| prime radical | C2×#41×C103 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C103 ◁ C2×#131×C103 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2,6,12,7,11,17,27,8,13,16,9,5,22,3,14)(4,10,20,26,21,25,29,32,18,24,30,23,19,31,15,28)(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);(1,3,5,16,8,17,7,6)(2,9,11,14,13,12,22,27)(4,15,19,30,18,29,21,20)(10,23,25,28,24,26,31,32);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,18)(9,23)(11,25)(12,26)(13,24)(14,28)(16,30)(17,29)(22,31)(27,32);(4,18)(10,24)(15,29)(19,21)(20,30)(23,26)(25,31)(28,32)9064 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26389);