Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C103 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C103×#1812 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #117 |
| commutator | C2×C4 | C23×C4×C103 |
| Frattini, cosocle | C22×C4 | C24×C103 |
| socle | C22×C103 | #380 |
| prime radical | C22×#41×C103 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C22×M16×C103 ◁ G | |
1346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 generators on 137 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(3,4)(5,13)(6,17,19,10)(7,20)(8,12)(9,15)(11,21)(14,27,29,25)(16,26)(18,24,33,32)(22,28,31,34)(23,30);(3,5,7,15,8,16,21,30)(4,9,11,13,12,23,20,26)(6,14,18,28,19,29,33,34)(10,22,24,25,17,31,32,27);(3,6)(4,10)(5,14)(7,18)(8,19)(9,22)(11,24)(12,17)(13,25)(15,28)(16,29)(20,32)(21,33)(23,31)(26,27)(30,34)6592 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 13658);