Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#26854 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| prime radical | C2×AGL(1,ℤ/8)×C103 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C22×C4×C103 ◁ G | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 generators on 127 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,21)(12,15)(13,23)(14,19)(16,20)(17,18)(22,24)(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);(1,3,5,4)(2,6,7,8)(9,11,15,22,16,23,14,18)(10,17,19,13,20,24,12,21);(1,4,5,3)(2,6,7,8)(9,12,14,20,16,19,15,10)(11,17,18,21,23,24,22,13);(2,7)(6,8)(9,13,15,24,16,21,14,17)(10,18,19,11,20,22,12,23)4120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 19842);