Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C103 · Schur multiplier (H2) = C23 |
| direct product | C103×#11602 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C103 | C2×D8 |
| commutator | C8 | C22×C8×C103 |
| Frattini, cosocle | C4×C8 | C23×C103 |
| socle | C2×C103 | C2×D4⋊C8 |
| prime radical | C2×#135×C103 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C103 ◁ C22×C8×C103 ◁ C2×C4×C8×C103 ◁ G | |
380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 3 generators on 135 points:
(1,2,5,9,6,11,21,26,7,12,22,27,20,10,4,18)(3,13,15,8,17,24,31,25,14,29,30,23,16,32,19,28)(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)(2,8)(4,19)(5,15)(6,17)(7,14)(9,24)(10,28)(11,25)(12,23)(13,18)(16,20)(21,31)(22,30)(26,29)(27,32);(3,14)(8,23)(13,29)(15,30)(16,17)(19,31)(24,32)(25,28)6592 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 4590);