Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C4 |
| direct product | C103×#7657 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C103 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C103 |
| Frattini, cosocle | C2×C42 | C23×C103 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C103 ◁ C43×C103 ◁ G | |
370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 26368 |
A minimal faithful permutation representation — 3 generators on 143 points:
(2,5,6,7)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,33)(30,36)(34,39)(37,40)5974 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 645);