Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C103 · Schur multiplier (H2) = C23 |
| direct product | C103×#13370 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C103 |
| prime radical | C23×C103 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C22×C4×C103 ◁ #405×C103 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,20)(12,26)(13,18)(14,31)(15,19)(16,22)(17,29)(21,36)(23,33)(24,40)(25,38)(27,35)(28,34)(30,37)(32,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(1,3,4,8)(2,5,6,7)(9,10)(11,21)(12,17)(13,31)(14,18)(15,19)(16,23)(20,36)(22,33)(24,32)(25,35)(26,29)(27,38)(28,37)(30,34)(39,40);(9,11,15,25)(10,16,19,32)(12,27,30,36)(13,24,14,23)(17,33,34,40)(18,35,31,21)(20,37,38,26)(22,28,39,29)4120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 6358);