Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C103 · Schur multiplier (H2) = C23×C4 |
| direct product | C103×#11229 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #117 |
| commutator | C22×C4 | C22×C4×C103 |
| Frattini, cosocle | C23×C4 | C23×C103 |
| socle | C22×C103 | #380 |
| prime radical | C24×C103 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C22×M16×C103 ◁ G | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 3 generators on 139 points:
(5,6)(7,16)(8,21,24,35)(9,25)(10,14)(11,15)(12,18)(13,23,28,22)(17,31,34,26)(19,27)(20,30)(29,33,36,32)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(1,2)(3,4)(5,7,9,18,11,20,10,19)(6,12,14,16,15,27,25,30)(8,17,22,32,24,34,23,33)(13,26,21,29,28,31,35,36);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,21,15,28,25,35)(7,17,18,32,20,34,19,33)(12,26,16,29,27,31,30,36)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 4217);