Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C25 |
| direct product | C2×C103×#1633 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C103 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C103 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C103 ◁ C2×C42×C103 ◁ G | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 26368 |
A minimal faithful permutation representation — 4 generators on 129 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(4,7)(5,9)(8,10)(11,12,15,18)(13,19,21,26)(14,22,24,17)(16,23,25,20);(5,9)(7,10)(11,13,15,21)(12,16,18,25)(14,23,24,20)(17,26,22,19);(3,4,6,8)(5,10,9,7)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 11495);