Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C2×C4 |
| direct product | C103×#13303 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | C2×#99 |
| commutator | C22×C4 | C22×C4×C103 |
| Frattini, cosocle | C22×Q8 | C23×C103 |
| prime radical | C23×C103 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C22×C4×C103 ◁ C2×C4×Q8×C103 ◁ G | |
396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 127 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,18,15,19,12,16)(11,20,22,13,17,21,24,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(2,5)(6,7)(9,11,21,18,15,17,20,16)(10,14,24,13,19,12,22,23);(1,3,4,8)(2,6,5,7)(9,12,15,14)(10,16,19,18)(11,22,17,24)(13,21,23,20)3811 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 6291);