Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C26 |
| direct product | C103×#10993 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #120 |
| commutator | C24 | C22×C4×C103 |
| Frattini, cosocle | C23×C4 | C23×C103 |
| socle | C22×C103 | #386 |
| prime radical | C2×#525×C103 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C23×C4×C103 ◁ G | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 3296 |
A minimal faithful permutation representation — 3 generators on 123 points:
(1,2)(3,4)(5,6,8,12,9,11,17,19)(7,13,15,20,14,10,16,18)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(2,4)(5,7)(6,10)(8,16)(9,14)(11,13)(12,20)(15,17)(18,19);(10,13)(18,20)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 3981);