Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#33750 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C103 |
| prime radical | C103×#2218 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C2×C42×C103 ◁ C2×C4×C8×C103 ◁ G | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 4 generators on 143 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,22)(13,27)(14,29)(15,19)(17,26)(18,24)(20,34)(21,33)(23,36)(25,37)(28,40)(30,35)(31,38)(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);(2,5)(6,8)(9,11,15,14)(10,16,19,29)(12,23,25,20)(13,21,28,24)(17,31,32,30)(18,27,33,40)(22,36,37,34)(26,38,39,35);(9,12,24,38,14,23,28,39,15,25,21,35,11,20,13,26)(10,17,27,34,16,30,33,37,19,32,40,36,29,31,18,22);(1,3,4,7)(2,6,5,8)(12,25)(17,32)(20,23)(22,37)(26,39)(30,31)(34,36)(35,38)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26738);