Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C24×C4 |
| direct product | C103×#17398 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C103 |
| Frattini, cosocle | C22×C4 | C24×C103 |
| prime radical | C25×C103 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C22×C42×C103 ◁ G | |
1034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 135 points:
(1,2,4,10,6,9,16,23)(3,8,13,22,11,21,24,32)(5,7,15,19,12,18,26,30)(14,17,28,29,25,20,27,31)(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)(3,11)(4,15)(8,20)(9,18)(13,27)(14,25)(16,26)(17,21)(22,32)(24,28)(29,31);(1,3,12,25)(2,8,18,20)(4,13,26,27)(5,14,6,11)(7,17,9,21)(10,22,30,31)(15,28,16,24)(19,29,23,32);(2,9)(7,18)(8,21)(10,23)(17,20)(19,30)(22,32)(29,31)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 10386);