Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C26 |
| direct product | C103×#17397 |
| 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 ◁ C24×C4×C103 ◁ G | |
1386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 4 generators on 127 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,14,17,12,16)(11,15,19,22,20,23,21,24)(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);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,23)(18,22);(9,12,14,13)(10,16,17,18)(11,19,20,21)(15,22,23,24);(10,17)(15,23)(16,18)(22,24)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 10385);