Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#25317 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C103 |
| prime radical | C22×D4×C103 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C2×C42×C103 ◁ G | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 4 generators on 143 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,18,13,17,27,34)(11,19,22,36,21,15,29,31)(12,23,26,33,25,35,28,16)(20,37,38,30,24,40,39,32)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(2,5)(6,7)(9,11,14,22,13,21,27,29)(10,15,18,31,17,19,34,36)(12,24,26,39,25,20,28,38)(16,32,23,37,33,30,35,40);(9,12,13,25)(10,16,17,33)(11,20,21,24)(14,28,27,26)(15,30,19,32)(18,35,34,23)(22,39,29,38)(31,37,36,40);(1,3,4,8)(2,6,5,7)(10,17)(15,19)(16,33)(18,34)(23,35)(30,32)(31,36)(37,40)4429 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 18305);