Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#23695 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C103 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| prime radical | C22×D4×C103 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C103 ◁ #376×C103 ◁ G | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 131 |
| transitive degree | μt(G) = 13184 |
A minimal faithful permutation representation — 4 generators on 131 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,18)(17,26)(19,21)(20,25)(23,27)(24,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131);(6,9)(10,12)(13,15,18,25)(14,20,16,22)(17,24,19,27)(21,28,26,23);(2,4)(13,16,18,14)(15,23,25,28)(17,26,19,21)(20,27,22,24);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,21,16,26)(15,24,25,27)(20,28,22,23)5356 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 16683);