Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C103 · Schur multiplier (H2) = C22×C42 |
| direct product | C103×#10814 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C103 | #117 |
| commutator | C2×C4 | C2×C42×C103 |
| Frattini, cosocle | C23×C4 | C23×C103 |
| socle | C22×C103 | #380 |
| prime radical | C24×C103 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C103 ◁ C22×M16×C103 ◁ G | |
658 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)(7,17)(8,22,24,29)(9,15)(10,14)(11,16)(12,19)(13,23,30,25)(18,26,33,36)(20,28)(21,27)(31,32,35,34)(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)(5,7,10,20,11,21,9,19)(6,12,15,27,16,28,14,17)(8,18,25,34,24,33,23,32)(13,26,22,35,30,36,29,31);(5,8,9,23,11,24,10,25)(6,13,14,29,16,30,15,22)(7,18,19,32,21,33,20,34)(12,26,17,31,28,36,27,35)6592 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 3802);