Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C25 |
| direct product | C2×C103×#2210 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C103 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C103 |
| prime radical | C22×D16×C103 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C103 ◁ C2×C42×C103 ◁ C2×C4×C8×C103 ◁ G | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 4 generators on 125 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(4,6)(7,8)(9,12)(10,13)(11,19)(14,18)(15,17)(16,20)(21,22);(3,4,5,6)(8,12)(13,20)(14,21)(15,19);(8,13)(10,18)(11,17)(12,20)(14,15)(16,22)(19,21)5974 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26563);