Finite Group DatabasebetaFinite Group Databasebeta| order | 26368 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C103 · Schur multiplier (H2) = C24 |
| direct product | C103×#33649 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C103 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C103 |
| prime radical | C24×C103 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C103 ◁ C23×C4×C103 ◁ C23×C8×C103 ◁ G | |
580 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)(3,6)(4,5)(7,8)(9,10,15,20)(11,21,24,38)(12,19,14,17)(13,26,29,34)(16,31,33,23)(18,25,36,28)(22,39,30,35)(27,37,40,32)(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)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,27)(14,30)(15,24)(17,32)(18,35)(19,37)(20,33)(21,26)(23,28)(25,31)(29,40)(34,38)(36,39);(9,12,15,14)(10,17,20,19)(11,22,24,30)(13,25,29,28)(16,32,33,37)(18,34,36,26)(21,35,38,39)(23,27,31,40)4738 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26368, 26637);