Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C25 |
| direct product | C103×#18520 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C103 |
| socle | C22×C103 | C2×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C2×C42×C103 ◁ G | |
990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 4 generators on 127 points:
(2,6)(3,4)(7,8)(9,10)(11,21)(12,24)(13,18)(14,19)(15,20)(16,22)(17,23)(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);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,13)(14,22)(15,23)(17,19)(18,21)(20,24);(1,3,5,4)(2,6,7,8)(9,12,15,22)(10,17,20,21)(11,13,23,14)(16,19,24,18);(9,13,15,14)(10,18,20,19)(11,12,23,22)(16,21,24,17)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 11508);